





































































/ 
















Geometrical 

Exercises 



By D. SANDS WRIGHT, A. M. 

Prof, of Mathematics in Iowa State Normal School 





v cofe, 

SEP 16 1892 ) 

377 l * * 


f 


CHICAGO 

A. FLANAGAN, Publisher 

1892 





Copyrigl.it, ,:8g2. 

A. FLANAGAN, Chicago 


































Contents 


PAGES 

Preface. 5-7 

Preliminary Definitions and Constructions. 9-11 

Straight Dines, Angles and Polygons. 12-24 

Circles. 25-39 

Proportion, Similar Polygons. 40-46 

Comparison and Measurement of Surfaces. 47-54 

Regular Polygons. The Area of the Circle. 55-60 

Planes. 61-65 

Solids.. 66-78 

Geometrical Formulae. 78 

Table of Formulae. 79 

Spherical Polygons. 80-82 

Supplementary Exercises. 83-84 

Table of the English Numerals. 84 

Vocabulary.85-104 

Greek and Datin Roots and Affixes. 104 




















PREFACE. 


Mo.dern methods of teaching geometry seek to encour¬ 
age, as far as possible, original investigation on the part of 
the learner. Such investigation is not likely to be secured 
from the student who finds in his text-book, on the page 
before him. the proposition and its demonstration. The 
object of this work is to present the exercises only, thus 
requiring the student to rely upon his own resources, or to 
seek assistance from other text-books, if he must. 

The work is not intended to supplant other text-books. 
Each student should have access to at least one standard 
geometry as a book of reference, to render aid when abso¬ 
lutely necessary, and to verify the correctness of his own 
demonstrations. 

The matter treated in the successive books and the or¬ 
der of presentation of the propositions, do not differ materi¬ 
ally from the corresponding treatment found in modern 
school geometries. The work is, therefore, adapted to use 
in connection with any text-book in this science in general 
use in the schools. 

The writer has endeavored, especially in the earlier 
books, to so place the exercises before the student as to 
lead him by inductive processes to discover his own demon, 
strations and solutions. 

The numbers in brackets throughout the work indicate 



vi 


PREFACE. 


the propositions usually demonstrated in the school geom¬ 
etries, and such as are likely to be of especial use to the 
student in his future mathematical studies. 

In the vocabulary will be found definitions of the terms 
involved in the science of elementary geometry. It is be¬ 
lieved that the etymologies given will assist the memory 
and add to the interest of the study. 

The special attention of teachers is invited to the exer¬ 
cises under the Pythagorean proposition and under the 
Similarity of Triangles, also to the treatment of the subjects 
of Symmetry (see Vocabulary) and the Theory of Limits. 

SUGGESTIONS TO THE STUDENT. 

Cultivate originality. 

The first essential to success is will power. Believe in 
yourself; resolve to succeed, and win. 

Begin at once to form the habit of honest and earnest 
endeavor to perform, without assistance, every exercise as 
it is presented. Time spent in such endeavor, even with¬ 
out success, is not wasted. 

The best mathematical discipline is that which is de¬ 
rived from the mastery of problems that tax to the utmost 
the energies of the mind. For disciplinary value, one hour 
of such study is worth a week of mechanical solutions of 
simple problems that require no mental effort. 

Seek aid from other text-books, if necessary. In doing 
so, note carefully the author’s diagrams and constructions, 
and endeavor, without further reading, to get from these a 
clue to his demonstration. Succeeding in this, the demon¬ 
stration still is half your own. 

When constructions are required, it is often useful to 
draw the figure as though the problem were already solved, 


PREFACE. 


vii 


and then by careful study of its parts to find the key to an 
original solution. 

Learn to think. 

ABBREVIATIONS AND SYMBOLS, 

D. Demonstrate. 

Hyp. Hypothesis. 

Q. E. D. Quod erat demonstrandum. 

Q. E. F. Quod erat faciendum. 

+ Plus. 

— Minus. 

X Multiplied by. 

-r- Divided by 
= Equals. 

> Angle. 

rt. > Right angle, 
rt. >s Right angles. 

1 Perpendicular. 

A Triangle. 

|| Parallel. 

> Greater than. 

Less than. 

Therefore, 




























. 























, 






















































































































































•- 









GEOMETRICAL EXERCISES. 9 

PRELIMINARY DEFINITIONS AND CONSTRUC¬ 
TIONS. 

1. Draw four lines. Mark them respectively a , b , c and 

d, making a>b, b>c, and c^>d. 

1. Construct a-\-b ; that is, construct a line equal to 

the sum of a and b. 

2. Construct a — b , a-\-b-\-c, a-\-b — c, a-\-{b — c ), ( a-\-b ) 

—c, (a-\-b) —( c-\-d ), #-|-(£— c)-\-d. 

3. Construct 2«, 2a-\-d, ^a-\-2b, $b —2 c, c+5d. 

2. Define Geometry, Magnitude, Extension. 

3. Can a magnitude have extension in one direction ? In 

two directions ? In three directions ? In more 
than three directions ? 

4. Define Point, Line, Surface, Solid. 

5. May a point be regarded as a line ; if so, what is its 

length ? May a point be regarded as a surface ; 
if so, what are its length and breadth ? May a 
point be regarded as a solid ; if so, what are its di¬ 
mensions ? 

6. Define a Line. 

1 . As a magnitude. 

2. As a path. 

7. Define a Straight Line. 

1 . As a magnitude. 

2 . As a path. 

3 . As a distance between two points. 

In like manner define (1) as a magnitude and (2) as a 
path. 

1 . A Curved Line. 

2 . A Surface. 

3 . A Plane Surface. 

4 . A Curved Surface. 


10 


GEOMETRICAL EXERCISES. 


8. Locate two points in a plane. Connect them by a 

straight line. How many such lines can be drawn? 
Call the connecting line c. Mark the distance be¬ 
tween the two points on a broken line a-\-b. The 
lines a and b being each of definite length, how 
many positions can a-\-b take ? 

9. Locate two points and connect them by a curved line 

d. If d is of definite form and length how many 
positions may it take? 

10. Can any two points in space be conceived as con¬ 

nected by a straight line ? 

11. Define Parallel Lines, Perpendicular Lines. 

12. Draw a Horizontal Line, a Vertical Line, an Oblique 

Line. 

13. Define a Horizontal Line, a Vertical Line, an Oblique 

Line. 

14. Draw two intersecting lines. 

Define an Angle, a Right Angle, a Straight Angle, an Oblique 
Angle, an Acute Angle, an Obtuse Angle, the Complement 
of an Angle, the Supplement of an Angle. 

15. How are angles measured? How many degrees in a 

right angle? In a straight angle? In three right 
angles? 

16. What is the supplement of an angle ot 12 0 ? What 

is the complement of an angle of 2 0 ? The comple¬ 
ment of an angle is 85°, what is its supplement? 

17. How many degrees in an angle that is twice its com¬ 

plement? In an angle that is three times its com¬ 
plement? 

18. The complement of an angle is of its supplement, 

how many degrees in the angle? 

19. Define Superposition. 


GEOMETRICAL, EXERCISES. 


11 


20. Define Proposition, Theorem, Problem, Axiom, Pos¬ 

tulate, Corollary, Lemma, Scholium, Formula. 
Axioms. The whole is greater than any of its parts. 

The whole is equal to the sum of all its parts. 

Things that are equal to the same thing or equal to each other. 
If equals be added to equals the results are equal. 

21. State axioms by completing the following pro¬ 

positions: 

If equals be subtracted from equals— 

If equals be multiplied by equals— 

If equals be divided by equals— 

Like parts of equals are— 

Equal powers of equals are— 

Equal roots of equals are— 

If unequals be— 

Added to equals— 

Subtracted from equals— 

Multiplied by equals— 

Divided by equals— 

Unequal powers of equals are— 

Unequal roots of equals are— 

If equals be— 

Added to unequals— 

Subtracted from unequals— 

Multiplied by unequals— 

Divided by unequals— 

Equal powers of unequals are— 

Equal roots of unequals are— 

All right angles are equal. 

Two straight lines cannot enclose a space. 

22. Postulates. 

1. Can any two points be connected by a straight line? 

2 . Can a straight line be produced to any distance? Can it be 

terminated at any point? 

3 . Can more than one straight line, liavina a given direction, be 

drawn through a given point? 


12 


geometrical exercises. 


BOOK I. 


STRAIGHT LINES, ANGLES AND POLYGONS. 

Define Adjacent Angles, Opposite or Vertical Angles, 
Supplementary-Adjacent Angles. 

Exercise [i.] 

If one straight line meet another the sum of the 
adjacent angles is equal to two right angles. 

Illustrative Demonstration . 


i. 


2 . 


c 


The Statement. If one straight line meet another the 
sum of the adjacent angles is equal to two right 
angles.' 

The Construction. 



Let the line A B meet the line C D at A. 
We are to prove that the angle C A B-\- 
the angle BAD is equal to two right 
right angles. 


The Demonstration . >E A C-Y^E A Z>=2rt.>s, since 
both angles are right angles. 

>B A C+> B A D= >E A C+- >E A D. 

Since things that are equal to the same thing are equal 
to each other? 

>B A C+>B A ZL=2rt.>s. Q. E D. 

1. To what is the sum of the angles formed by drawing 

any number of lines on the same side of a straight 
line, to a point in that line, equal? 

2. If from a point a number of straight lines be drawn, 

to what is the sum of the angles about the point 
equal ? 




geometrical exercises. 


13 


3. Can the exterior sides of two supplementary adja¬ 
cent angles be a broken line ? 

[2.] If two straight lines intersect, the vertical angles 
are equal. 

ILLUSTRATIVE DEMONSTRATION. 

1. The Statement. 

If two straight lines intersect, the vertical angles are 
equal. 

2. The Construction. 

A D 

Let the straight lines A B and C D intersect 
at O. 

We are to prove that <i=<i', and that 
< 2 =< 2 '. 

The Demonstration. 

<i + <2—2 rt. <s, being supplementary adjacent 
angles. 

<2 +<i'=2 rt.<s, being supplementary adjacent 
angles. 

<i-f >2—>2+>i'; things that are equal to the 
same thing are equal to each other. 

<i=<i'. If equals be subtracted from equals, the 
results are equal. 

In like manner it may be proven that >2=<2'. 

.\ The vertical angles 1, 1' and 2, 2', are respectively 
equal. Q. E. D. 

I. Two straight lines intersect at a point. One of the 
angles formed is 20°. Find the number of degrees 
in each of the remaining angles. 



GEOMETRICAL EXERCISES. 


2. n lines are drawn to a common point, now many 

angles are formed ? 

3. n lines intersect at a common point, liow many angles 

are formed? 

4. Five angles are formed by drawing lines to a common 

point, so that their magnitudes are in the propor¬ 
tion of 1, 2, 3, 4 and 5 respectively, how many 
degrees in each angle ? 

5. Two straight lines A B and C D intersect at O, and 

O M and O N are drawn respectively bisecting the 
pair of vertical angles A O D and COB. Is M O N 
a straight or a broken line? D. 

6. Two supplementary adjacent angles having their 

Vertices at O are bisected by the lines O M and O 
N. Is the angle MON, right, acute, or ob¬ 
tuse ? D. 

[3.] If a perpendicular be erected at the middle point of a 
straight line. 

1 . Can any point in the perpendicular be unequally distant 

from the extremities of the straight line? D. 

2 . Can any point without the perpendicular be equally distant 

from the extremities of the straight line. 

[4.] How many perpendiculars can be drawn from a given 
point to a given straight line? D. 

[5.] The sum of two lines drawn from a point to the ex¬ 
tremities of a straight line, is greater than the sum 
of any two other lines similarly drawn but in¬ 
cluded by them. 

[6.] If from a point in a perpendicular to a given line, 
two unequal oblique lines be drawn, the line meet¬ 
ing the given line at the point more remote from 
the foot of the perpendicular is the greater. 


geometrical exercises. 


15 


1. How many equal oblique lines can be drawn from a given 

point to a given straight line ? 

2 . How would you find the shortest distance from a point to 

a straight line? 

3 . From a point in a given straight line how many perpen¬ 

diculars can be erected? 

7. How many lines can be drawn through a point par¬ 

allel to a given line ? 

1. How many lines can be drawn parallel to a 

given line and at a given distance from it? 

2. If two straight lines are each parallel to a 

third straight line, are they parallel to each 
other ? D. 

3. If two straight lines are each perpendicular to 

a third straight line, are they parallel to each 
other ? D. 

Define the Locus of a Point. 

8. What is the locus : 

1. Of a point equidistant from two given points? 

2. Of a point at a given distance from a given 

line? 

3. Of a point equidistant from two parallel 

straight lines? 

9. The straight line connecting two points is a due 

north and south line, a moving point passes be¬ 
tween them: 

1. By what route must it pass to be always equi¬ 

distant from A and B? 

2. By what route so as to be always nearer A 

than B? 

3. By what course so as to be equidistant from A 

and B but once ? 


16 


geometrical exercises. 


What is a transversal ? 

Draw two parallel straight lines and a transversal 
cutting them. Mark and indicate the pairs 
formed of: 

1. Alternate interior angles. 

2 . Alternate exterior angles. 

3 . Exterior interior angles. 

4 . Opposite exterior angles. 

5 . Interior angles on the same side of the transversal. 

[10 ] If two parallel straight lines are cut by a trans¬ 
versal, the alternate interior angles are equal 

[11.] State and prove the conversely. 

[12.] If two parallel straight lines are cut by a transver¬ 
sal, the exterior interior angles are equal. 

[13.] State and prove the conversely. 

[14.] If two parallel straight lines are cut by a trans¬ 
versal, the sum of the interior angles on the same 
side of the transversal is equal to two right angles. 

[15.] State and prove the conversely. 

[16.] What relation subsists between two angles whose 
sides are respectively parallel and lying- 

1 . In the same direction? D. 

2 . In opposite directions ? D. 

3 . One side in the same direction, and the other in the op¬ 

posite direction from the vertex ? D. 

What relation subsists between two angles whose sides 
are respectively perpendicular and lying: 

1 . In the same direction ? D. 

2 . In opposite directions ? D. 

3 . One pair of sides in the same direction and the other in 

the opposite direction from the vertex ? I>. 

What is a Polygon? 

Give the name ot a polygon of 5 sides; ot. 6 sides; of 7; 


GEOMETRICAL EXERCISES. 


17 


of 8 ; of 9 ; of 10 ; of n; of 12 ; of 13 ; of 14 ; of 15 ; 
of 16 . 

1 . What is a Triangle. 

1 . Construct a triangle having no two sides equal. 
Having but two sides equal. 

Having its three sides equal 

Remark .—The last exercise above and other like exercises below 
can only be performed approximately at this stage in the 
learner’s progress. The processes of exact constructions can 
not be employed until the relations of the.circle have been 
investigated. 

2 . Construct a triangle. 

Containing a right angle. 

Containing an obtuse angle. 

Having all its angles acute. 

3 . Construct a quadrilateral 

Having no two sides parallel. 

Having but two sides parallel. 

Having its opposite sides parallel. 

4 . Construct an oblique angled parallelogram. 

Having its adjacent sides unequal. 

Having all its sides equal. 

5 . Construct a right angled parallelogram. 

Having its adjacent sides unequal. 

Having all its sides equal. 

Define the terms in the outlines below. 


As to sides 



Triangle 


As to angles 


f Right ( 
{ Oblique ( 


Acute 

Obtuse 



18 


geometrical exercises. 


Quadri 

lateral 


' Trapezium 

^ (Isoceles 

Tropezoid | g calene 


f Rhomboid—Rhombus 
Parallelogram j Rectangle j Ohlong 

f 17.] The sum of the three cngles of a triangle is equal to 
two right angles. 


I 


Demonstrate: 

1. By drawing a line from the vertex of an angle parallel to 

the side opposite that angle. 

2. By drawing a line through the vertex parallel to the base. 

1. Can a triangle have more than one obtuse angle? More than 

one right angle? 

2. Two angles of a triangle are respectively 20° and 30°, liow 

many degrees in the third angle? 

3. If two angles of a triangle are known, how may the third angle 

be found ? 

4. If two triangles have two angles of the one respectively equal 

to two angles of another triangle, what is true of the third 


angles ? 

5. Compare an exterior angle of a triangle with the sum of the 

remaining angles. 

6. To what is the sum of the acute angles of a right triangle 
equal ? 

7. What is the relation of one acute angle to the other acute 
angle of a right triangle ? 

8. The three angles of a triangle are in the proportion of 1, 3 and 

5. How many degrees in each angle of the triangle ? 

9. Construct the triangle ABC \ produce the side A B at B, also 

the side B C at C, and the side A C at A. Find the sum 01 
the exterior angles so formed. 

10. Find the sum of the interior angles of a quadrilateral. 

Suggestion. Construct a quadrilateral and draw one 

of its diagonals. 

11. Find the sum of the exterior angles of a quadrilateial. 

12. Find the sum of th e interior a ngles of a pentagon. 






geometrical exercises 


19 


13. Find the sum of the interior angles of a hexagon. 

14. Find the sum of the interior angles of a polygon of n sides. 

15. Find the sum of the exterior aflgles of a pentagon. 

16. Find the sum of the exterior angles of a hexagon. 

17. Find the sum of the exterior angles of a polygon of n sides. 

18. The angles of a quadrilateral are in the proportion of 3, 5, 7 
and 9. How many degrees in each angle? 

19. The sum of the interior anglesof a polygon is 28 right angles. 
How many sides has the polygon ? 

20. The sum of the interior angles of a polygon is double the sum 
of its exterior angles. How many sides has the polygon ? 
What is meant by Equal Figures? 

[18]. Two triangles are equal, if two sides and the included 
angle of the one are equal respectively to two sides 
and the included angle of the other. 

1. Can equality be affirmed if two triangles having two angles 

and the included side of the one equal respectively to two 
angles and the included side of the other? D. 

2. Can equality be affirmed of two triangles: 

1. Having the three sides of the one equal respectively to 

the three sides of the other? D. 

2. Having the three angles of the one equal respectively to 

to the three angles of the other? 

3. Having two angles and a side of the one equal respec¬ 

tively to two angles and a side of the other? 

4. Having two sides and an angle of the one equal respec¬ 

tively to two sides and an angle of the other? 

3. Can equality be affirmed of two right triangles, which have re¬ 

spectively equal angles, each to each: 

1. The hypotenuse and a leg? 

2. The two legs ? 

3. The hypotenuse and an acute angle ? 

4. A leg and an acute angle ? 

5. The two acute angles ? 

[19.] In an isosceles triangle, the angles opposite the 
equal sides are equal. 


4 


20 geometrical exercises. 

Demonstrate: 

1. By drawing a perpendicular from the vertex to the base. 

2. By drawing a line from the vertex to the middle of the 

base. 

3. By extending to the base the bisector of the vertical angle. 

[20.] State and prove the conversely of nineteen. 

1. How many degrees in each angle of an equilateral tri¬ 

angle ? 

2. How many degrees in one of the acute angles of an isos¬ 

celes right triangle ? 

3. If two sides of a triangle are equal, and any one of its 

angles is 6o°, the triangle is equilateral. 

4. Are all equiangular triangles equilateral ? D. 

5. Are all equilateral triangles equangular? D. 

[21.] If two triangles have two sides of the one equal re¬ 
spectively to two sides of the other, and the in¬ 
cluded angle of the first greater than the included 
angle of the second, the third side of the first is 
greater than the third side of the second. 

[22.] State and prove the conversely. 

[23.] Of two sides of a triangle the greater side is op¬ 
posite the greater angle. 

[24.] State and prove the conversely. 

25. Bisect an angle and prove : 

1 . That any point in the bisecter is equally distant from the 

sides of the angle ? 

2 . That any point without the bisecter is unequally distant 

from the sides of the angle ? 

26. Join the middle points of the sides of a triangle and 

prove that the four triangles formed are equal. 

27. The bisectors of the three angles of a triangle meet 

in a point. 

28. The perpendiculars erected at the middle points of 

the sides of a triangle meet in a point. 


GEOMETRICAL EXERCISES. 


21 


Under what conditions will the point of meeting 

be ? 

1. Within the triangle? 

2. Without the triangle ? 

3. In the perimeter of the triangle ? 

29. The perpendiculars from the vertices of a triangle to 

the opposite sides meet in a point. 

Under what conditions will the point of meeting 

be ? 

1. Within the triangle ? 

2. Without the triangle ? 

3. In the perimeter of the triangle ? 

30. The bisector of the exterior angle at the vertex of an 

isosceles triangle is parallel to the base. 

31. Test the equality of the perpendiculars to the equal 

sides of an isosceles triangle from the angles op¬ 
posite. 

1. In an acute triangle. 

2. I 11 an obtuse triangle. 

32. Test the equality of the bisecters of the equal angles 

of an isosceles triangle produced until they meet 
the equal sides. D. 

33. Test the equality of the medians drawn from the equal 

angles of an isosceles triangle to the opposite 
sides. D. 

34. Test the equality of the triangles formed by drawing 

the diagonal of a rectangle. D. 

1. Draw both diagonals of a rectangle and test the equality 

of the pairs of triangles formed. 

2. What relation subsists? 

1. Between the opposite angles of a parallelogram. 

2. Between the adjacent angles of a parallelogram. 

[35.] The diagonals of a parallelogram bisect each other. 


4 


22 


geometrical exercises. 

36. What relation subsists between the diagonals: 

1. Of a rectangle? 

2. Of a rhombus? 

3. Of an isosceles trapezoid ? 

37. Draw quadrilaterals as indicated below, and deter¬ 

mine which of them are parallelograms. 

1. The diagonals bisecting each other. 

2. The opposite sides equal. 

3. A pair of opposite sides equal and parallel. 

4. Two pairs of adjacent sides equal. 

38. In an isosceles trapezoid, what relation subsists 

between: 

1. The opposite angles ? 

2. The two angles adjacent to one of the parallel sides? 

3. The two angles adjacent to one of the non-parallel sides? 

39. If from any point within a triangle, lines be drawn 

to its vertices, their sum is greater than one-half 
the perimeter of the triangle. 

40. Find the sum of the five angles formed by producing 

all the sides of a pentagon till they meet. 

[41.] If a series of parallel lines cut off equal parts on any 
transversal, they will cut off equal parts on every 
transversal. 

42. If three or more straight lines cutoff equal parts on 
any transversal they are parallel. 

[43.] Join the median (see glossary) of a trapezoid, and 
prove: 

1. That it is parallel to the parallel sides of the trapezoid. 

2. That it is equal to half their sum, 

44. Join the middle points of the sides of a triangle by a 
straight line and prove: 

1. That it is parallel to the base of the triangle. 

2 . That it is equal to half the base. 


GEOMETRICAL EXERCISES. 


23 


45. Join the middle points of the adjacent sides of a 

trapezium and prove that the quadrilateral formed 
is a parallelogram. 

46. What form of the parallelogram is produced b) r join¬ 

ing the middle points of the adjacent sides ? 

1. Of a sqnare ? D. 

2. Of a rhombus ? D. 

3. Of an oblong ? D. 

4. Of an isosceles trapezoid ? D. 

5. Of a trapezium having two opposite pairs of adjacent sides 

equal ? D. 

47. P and Q are respectively the middle points of the 

opposite sides A B and C D of the parallelogram 
ABC B. Prove that the diagonal A D is tri¬ 
sected by the lines P C and B Q. 

48. The medians of a triangle meet in a point, and one- 

tliird of each median is cut off at the point of 
meeting. 

Suggestion .—Complete the parallelogram by drawing from the 
extremities of one side parallels to the other two sides. 

49. Any straight line (the diagonals excepted) drawn 

through the point of intersection of the diagonals 
of a parallelogram, divides the parallelogram into 
two equal trapezoids. 

50. On the diagonal A D of the square A B C D, A E is 

taken equal to A B, and from the point E, E F is 
drawn perpendicular to A D, prove that E D=E F 
=B F. • 

51. A quadrilateral is equilateral if its diagonals bisect 

each other at right angles. 

52. The middle point of the hypotenuse of a right tri¬ 

angle is equally distant from the three vertices. 


4 


2 ^ 


geometrical exercises. 

5^ The bisectors of two adjacent angles of a parallelo¬ 
gram meet at right an glee. 

54. If one angle of aright triangle is 6o°, the hypotenuse 

is double the shorter side. 

55. Find the number of degrees: 

1 . Tn each angle of a equiangular hexagon. 

2 . In one of the exterior angles of an equiangular octagon. 

3 . In the angle formed by the bisectors of the angles A and 

C of the equiangular hexagon A B C D E F. 

56. How many diagonals can be drawn in a polygon : 

1 . Of 4 sides? 

2 . Of 6 sides ? 

3 . Of n sides? 


« 


geometrical exercises 


25 


BOOK II. 

CIRCLES. 

1. Define a Circle, a Semi-circle, a Quadrant. 

May a circle be defined as a polygon? 

2. Define Circumference, Semi-circumference, Arc. 

How many degrees in a circumference; in a semi- 
circumference; in a quadrant? 

3. Define Diameter, Radius, Chord. 

Can you draw in the same circle two diameters of 
unequal length? 

I11 a given circle, can you draw a chord longer 
than a diameter? 

4. Define Segment, Sector. 

How many segments are constructed by drawing a 
chord ? 

Divide a circle into two equal segments. 

Pass two diameters through a circle. Into how 
many sectors is it divided ? 

Draw two radii in a circle. 

How many sectors have you formed ? 

Cut off a part of a given circle that shall be both a 
segment and a sector. 

Draw two equal chords in a circle and prove that 
they are equally distant from the center. * 

Draw two unequal chords in a circle and determine 
which is the more remote from the center. 

Define Central Angles. 


26 


geometrical exercises. 


[i]. In the same circle or in equal circles : 

1. Construct two equal central angles and test the equality of the 

intercepted arcs. 

2 . Construct two equal arcs and test the equality of the sub¬ 

tended central angles. 

3 . Draw two equal chords and test the equality of the inter¬ 

cepted arcs. 

4 . Construct two equal arcs and test the equality of the sub¬ 

tended chords. 

Define Secant, Tangent. 

Define Inscribed Angle, Inscribed Polygon, Circum¬ 
scribed Polygon. 

When is a circle inscribed in a polygon ? When is a 
circle circumscribed about a polygon ? 

What are Concentric Circles ? 

[2.] A radius perpendicular to a chord bisects the chord 
and the arc subtended by it. 

Show that the following lines will pass through 
the center of the circle . 

1 . The perpendicular erected at the middle point of any 

chord. 

2 . The straight line passing through the middle points of a 

chord and its subtended arc. 

3 . The perpendiculars to the middle points of the sides of an 

inscribed quadrilateral. 

[3.] A tangent to a circle meets at right angles the radius 
drawn to the point of contact. 

[4.] State and prove the conversely. 

What relation subsists between : 

1. Two tangents drawn to a circle from a point without the 

cincumference ? 

2 . The two tangents drawn through the extremities of a 

diameter ? 


GEOMETRICAL EXERCISES. 


27 


LIMITS. 

1. When are lines commensurable ? What is meant by 

a unit of measure ? When are two lines incom¬ 
mensurable ? 

2. Two lines are respectively 2 ft. and 3 ft. in length, are 

they commensurable lines ? What is the unit of 
measure ? What is their ratio? Name a unit that 
will measure two lines respectively one inch and 
seven-inches in length. 

3. Suppose two straight lines A and B to be in the ratio 

A 

of y 7 2: i, then =1/2=1.4142'-+-. 

B 

By how much is the expression 1.4 nearer the correct 
numerical result than the 1 unit at the left of the 
separatrix? To within what fraction does 1.4 ex¬ 
press the exact ratio? 1.41 ? 1.414? 1.4142? 

1.41421 ? By this process can the exact value 
be obtained ? Can an expression within zero 
of the correct result be obtained ? In the process of 
the above approximation, what is the relative 
value of the units employed ? 

4. Suppose two straight lines, A and B, to be incommen¬ 

surable, and that a measure can be applied to B 
exactly n times and to A m times with a remain¬ 
der. Does m express the ratio of A to B ? Suppose 
n 

each of the n divisions of B divided into n equal 
parts, and one of the n divisions of B applied to A, 
making more than nm divisions with a remainder. 
Call the result m r plus a remainder. Which is the 

1)1 Vi' 

nearer approximation to the ratio of A to B, or - 

n 11 

Again, suppose B divided into if divisions and A into 
m"-\- a remainder, and continue the process, ob- 


28 


GEOMETRICAL EXERCISES. 


771 771 771 

taining successively ——r-etc. Can the 

71 TV 7t d 

exact ratio be obtained by this method? To within 
what limit can it be approximated? 

5. Upon a straight line a b a variable point is moving 

from a towards b. During the first minute it moves 
half the distance from a to b , the second minute 
half the remaining distance, the third minute half 
the remaining distance, and so on. When will v 
reach b ? Is a b a constant or a variable? The line 
a v ? The line v ft In what respect does the variable 
a v differ from the variable v ft Can v have 
such a position that a v shall equal a ft. 

6. Define Constant, Variable, Increasing Variable, De¬ 

creasing Variable, Limit, Superior Limit, In¬ 
ferior Limit. 

7. Can the sum of two variables be a constant? Can the 

product of two variables be a constant? 

8. Determine whether each of the results indicated below 

is a constant or a variable. 

1. The sum of a constant and a variable. 

2. The difference of a constant and a variable. 

3. The product of a constant and a variable. 

4. The ratio between a constant and a variable. 

5. The sum of two increasing variables. 

6. The sum of two decreasing variables. 

7. The sum of an increasing and a decreasing variable? 

8. The difference of two increasing variables. 

9. The difference of two decreasing variables. 

10. The difference of an increasing and a decreasing variable. 

11. The product of two increasing variables. 

12. The product of two decreasing variables. 

13. The product of an increasing and a decreasing variable? 



GEOMETRICAL EXERCISES. 


29 


9. What is the superior limit of the sum of two increas¬ 

ing variables whose limits are equal? The superior 
limit of their product? 

10. Show: 

1. That if two variables, either both increasing or both de¬ 

creasing, are always equal, their limits are equal. 

2 . That if the ratio of two variables is a constant, the ratio 

of their limits has the same ratio as that of the variables. 

[5]. In the same circle or in equal circles, equal central 
angles have the same ratio as their intercepted 
arcs. 

1. When the arcs are commensurable. 

1. When the arcs are incommensurable. 

Illustrativq Dcmonstration . 

The Statement: In the same circle or in equal circles, 
equal'central angles have the same ratio as their 
Case 2. When the arcs are in- 


The Construction. In the circle 
whose center is O, let the arcs 
A B and B C intercepted by the 
central angles A OB and BOC 
be incommensurable. 

„ , AB' >AOB 

To prove that gc = >BbC 

Divide B C into a number of equal parts. One of these 
parts may be applied to A B a certain number of 
times with a remainder D B. 


intercepted arcs, 
commensurable. 


A 






30 


GEOMETRICAL EXERCISES. 


The Demonstration: Since the arcs A D and B C are com- 


w AD >AOD 
mensurable, B C = >B Q c 

If the unit of measure be diminished indefinitely, 
the remainder D B will be likewise diminished, the 


arc A D will approach the arc A B at its limit, and 
the > A O D will approach the > A O B as its limit. 
But whatever the position of D, the variable ratio 
of A D to B C remains equal to the variable ratio of 

> A O D Since, if two ratios are always equal 

> B O C 

+1 . r « AB >AOB 

their limits are equal Q. E . D. 

[6] . Parallel lines intercept equal arcs of a circumference: 

1. When tlie parallels are tangents 

2 . When the parallels are a secant and tangent. 

3 . When the parallels are secants. 

If two diameters of a circle meet at right angles, how 
many degrees in each of the intercepted arcs ? If 
one of the right central angles be bisected, how 
many degrees in each of the intercepted arcs? If 
two circles with unequal radii have a common 
center at the vertex of a given angle, will the 
number of degrees in each of the arcs intercepted 
by the sides of the angle be the same? 

[7] . An inscribed angle is measured by one half the in¬ 

tercepted arc. 

Case 1 . When one side of the angle is a diameter. 

Suggestion: Through the center of the circle draw a 

diameter parallel to the shorter side. 

Case 2 . When the center is within the angle. 

Case 3. When the center is without the angle. 

1 . What is the measure of an angle inscribed in a semicircle? 


GEOMETRICAL EXERCISES. 


31 


2. What is true: 

1. Of all angles inscribed in the same segment? 

2 . Of an angle inscribed in a segment less than a semi 

circle? 

3. Of an angle inscribed in a segment greater than a semi¬ 

circle. 

[8] . Find tlie measure of an angle formed by a tangent 

and a chord? See Suggestion under 12, Case 1. 

[9] . Find the measure of the angle formed by two chords 

which intersect within the circle. See Suggestion 
under 12, Case 1. 

[10] . Find in terms of the intercepted arcs the measure of 

the angle formed: 

1. By two secants which meet without the circle. 

2. By two tangents which meet without the circle. 

3. By a secant and a tangent which meet without the circle. 
In each of the last three exercises see Suggestion under 

12, Case 1. 

11. What form of polygon is produced by drawing tan¬ 

gents through the vertices of an inscribed : 

1 . Equilateral triangle? D. 

2 . Isosceles triangle? D. 

3 . Square? D. 

4 . Rectangle? D. 

12. A circle being given, to find the ratio: 

1 . Of the circumscribed to the inscribed square. 

2 . Of the circumscribed to the inscribed equilateral triangle. 

13. A triangle is circumscribed about a circle by drawing 

tangents through the vertices of an inscribed isos¬ 
celes triangle. The vertical angle of the inscribed 
triangle is an angle of io°; how many degrees in 
each angle of the circumscribed triangle ? 

14. The angle at the vertex of a circumscribed isosceles 

triangle is 24 0 . How many degrees in each angle 


32 


GEOMETRICAL, EXERCISES 


of the inscribed triangle formed by straight lines 
joining the points of tangency? 

15. A circle tangent internally to another circle has for 
its diameter the radius of the larger circle. Prove 
that every chord of the larger circle drawn from 
the point of tangency is bisected by the circumfer¬ 
ence of the smaller circle. 


PROBLEMS. 


[16]. To erect a perpendicular to a given straight line at 
a given point in the line. 

Case 1. When the point is within the line. 

Case 2. When the point is at the extremity of the line. 


ILLUSTRATIVE SOLUTION. 


I. The Statement. To erect a perpendicular to a given 
straight line at a given point in the line. 

Case 1 . When the point is within the line. 



2. The Construction. Let A B 
be the given line and C the 
given point. It is required 
to erect a perpendicular to 
A B at C. 


From the point C with any radius describe an arc cutting 
A B at F and G. From F and G with any radius greater 
than one-half of F G describe arcs intersecting at some point 
as E, draw the straight line E C, which is the perpendicular 
required. 

3. The Demonstrate. E and C are two points equally dis¬ 
tant from the extremities of the straight line F G, 




GEOMETRICAL EXERCISES. 


33 


and therefore determine a perpendicular to the line 
at the point C. Q. E. F. 

[17] . From a point without a straight line to draw a per¬ 

pendicular to that line. 

[18] . To bisect : 

1. A given straight line. 

2. A given arc. 

3. A given angle. 

[19] . At a given point in a straight line to construct an 

angle equal to a given angle. 

[20] . Through a given point to draw a line parallel to a 

given line. 

[21] . Two angles of a triangle being given, to find the 

third angle. 

LOCI. 

22. Find the locus of a point at a given distance: 

1 . From a given point. 

2. From a given straight line. 

3 . From a given circumference. 

4 . From the perimeter of a given triangle. 

23. Find the locus of a point equidistant: 

1 . From two given points. 

2 . From two given parallel straight lines. 

3. From two given intersecting straight lines. 

4. From two. given concentric circumferences. 

24. Given the base a b of the triangle a b c and the 

length of the side a c , to find the locus of the 
vertex c. 

25. Given two sides of a triangle and the vertex of the 

angle they include, to find the loci of the other 
vertices. 

26. Given the hypotenuse of a right triangle, to find 

the locus of the vertex of the right angle. 


34 geometrical exercises. 

27. Find the locus of the centre of a circle that with a 

given radius shall touch : 

1. A given straight line. 

2. A given circumference. 

28. Find the locus of the middle point of a given chord 

drawn in a given circle. 

29. Find the locus of the center of a circle which shall 

touch : 

1. A given straight line at a given point. 

2. A given circumference at a given point. 

30. Given the diagonals of a parallelogram and their 

point of intersection, to find the loci of the ver¬ 
tices of the parallelogram. 

31. Given the three medians of a triangle and their point 

of intersection, to find the loci of the vertices of 
the triangle. 

32. Given the side of an equilateral triangle, to construct 

the triangle. 

33. To construct an angle of 6o°, of 30°, of 45 0 , of 120°, 

of 75 0 , of 150°, of 135 0 . 

[34] . Given the .three sides of a triangle, to construct the 

triangle. 

[35] . Given two sides and the included angle of a tri¬ 

angle, to construct the triangle. 

[36] . Given two angles and the included side of a tri¬ 

angle, to construct the triangle. 

[37] . Given two sides of a triangle and the angle opposite 

one of them, to construct the triangle 
Case 1. When the side opposite is the shorter of the given sides, 
and the given angle is acute. 

Case 2 . When the side opposite is the longer of the given sides, 
and the given angle 


geometrical exercises. 


35 


1. Is acute. 

2 . Is a right angle. 

3 . Is obtuse. 

[38] . To construct a right triangle, having given : 

1 . The two legs. 

2 . The hypotenuse and a leg. 

3 . The hypotenuse and an acute angle. 

4 . A leg and an acute angle. 

[39] . Given two sides and the included angle of a paral¬ 

lelogram, to construct the parallelogram. 

[40] . To pass a circle through three points not in the same 

straight line. 

[41] . To inscribe a circle in a given triangle. 

[42] . To draw a tangent to a circle : 

1 . Through a point in the circumference. 

2 . From a point without the circle. 

[43] . Upon a given straight line to describe a segment that 

shall contain a given angle. 

[44] . To find the ratio of two commensurable straight 

lines. 

45. To trisect: 

1. A given semi-circumference. 

2. A given quadrant. 

46. To construct an isosceles right triangle, the hypote¬ 

nuse being given. 

47. To construct an isosceles triangle, having given : 

1. Its base and altitude. 

2. Its altitude and one of the equal sides. 

3 . Its altitude and the angle at the vertex. 

4 . Its base and the angle at the vertex. 

5 . Its perimeter and the angle at the vertex. 

6 . The base and the vertical angle. 

48. To construct a triangle, having given : 

1 . The base, the altitude and an angle at the base. 


36 


geometrical exercises. 


2 . The base, the altitude and one side. 

3. The altitude and the two angles at the base. 

4. The base, the angle opposite the base, and the altitude. 

5. The base, the angle opposite the base, and the median 

line to the base. 

6 . The perimeter and the angles at the base. 

7. Two sides and the median drawn from their point of inter¬ 

section. 

8 . The three medians. 

9. The base, an angle at the base, and the sum of the other 

two sides. 

10. The base, an angle at the base, and the difference of the 
other two sides. 

49. To construct a square, having given : 

1 . The sum of the side and the diagonal. 

2 . The difference of the side and the diagonal. 

50. To construct a parallelogram, having given : 

1 . The diagonals and the angle at which they intersect. 

2 . The diagonals and the angle that one of them makes with 

a side. 

3 . The sides and one diagonal. 

4 . The diagonals and one side. 

51. To construct a rhombus, having given : 

1. The diagonals. 

2 . A side and an angle. 

3 . A diagonal and the angle from which it is drawn. 

4 . A diagonal and the angle opposite. 

52. To find a point at a given distance from a point A and : 

1 . At a given distance from a given point. 

2 . At a given distance from a given straight line. 

3 . At a given distance from a given circumference. 

4 . Equidistant from two parallel straight lines. 

5 . Equidistant from two intersecting straight lines. 

6 . Equidistant from two concentric circumferences. 

53. To describe a circumference that shall touch : 

1. A given line at a given point A, and pass through a given 
point B. 


geometrical exercises. 


37 


SuggEstion .— Find the point of intersection of two loci. 

2 . A given line A B at a given point P, and touch a given 

line C D. 

3 . Two given lines and pass through a given point. 

a. When the lines are parallel. 

b. When the lines are convergent. 

4 . Two parallel lines and a transversal. 

5 . A given circumference at a given point A, and pass through 

a given point B. 

6 . The sides of a given angle and touch one of the sides at 

a given point. 

7 . A given circumference at a point A, and touch a given 

straight line. 

54. To find a point equidistant from two given points 

and also equidistant: 

1. From two parallel straight lines. 

2 . From two intersecting straight lines. 

3 . From two concentric circumferences. 

55. To draw the shortest chord that can be passed 

through a given point within a circle. 

56. To find the locus of the middle point of all chords: 

1 . That can be passed through a given point within a circle. 

2 . That when produced will pass through a given point with¬ 

out a circle. 

57. The sums of the opposite sides of a circumscribed 

quadrilateral are equal. 

58. If from the vertices of a circumscribed quadrilateral 

straight lines be drawn to the center of the circle, 
the sum of each pair of opposite central angles 
formed is equal to two right angles. 

59. If on the diameter of a circle as a base an equilateral 

triangle be constructed, the circumference will 
bisect the sides of the triangle. 

60. To draw a common tangent to two circles : 

1 . Externally. 


38 


GEOMETRICAL EXERCISES. 


61. To inscribe in a given rhombus : 

1 . A square. 

2. A circle. 

62. Divide a given triangle: 

1. Into four equal triangles. 

2. Into nine equal triangles. 

3. Deduce a general method for dividing a given triangle 

into n 2 equal triangles. 

63. Divide a given parallelogram : 

1 . Into four equal parallelograms. 

2 . Into nine equal parallelograms. 

3 . Into eighteen equal triangles. 

64. Draw a line that shall have a given length, be par¬ 

allel to the base of a triangle, and be terminated 
by the sides (or sides produced) of the triangle. 

1 . When the given line is shorter than the base. 

2 . When the given line is longer than the base. 

65. To draw a line that shall have a given length, pass 

through a given point and have its extremities in 
two parallel lines. 

66 . Through a given point between two concentric cir¬ 

cumferences to draw a straight line that shall be 
terminated by the circumferences and bisected by 
the point. 

67. To inscribe a circle in a given sector. 

63 . What form of the quadrilateral is produced by the 
bisectors of the angles : 

1. Of a rhomboid? D. 

2. Of an oblong? D. 

69. If in the sides A B, B C, C D, and D A of the paral¬ 
lelogram A B C D there be taken respectively 
A K-B F=C G=D H, and the points of intersec- 


GEOMETRICAL EXERCISES, 


39 


tion be joined by straight lines, what form of the 
quadrilateral is the figures EFGH? 

1. If A B C D is a square? 

2. An oblong ? 

3. A rhomboid.? 

70. To trisect a given straight line. (Two methods.) 

1. The diagonal of a parallelogram, 

2. The median of a triangle. 

71. Given the base of a triangle and the angle opposite 

the base, to find the locus of the vertex of the given 
angle. 

72. Given in position the middle points of the sides of 

a triangle to construct the triangle. 

73. Given the middle points of two sides of an equilat¬ 

eral triangle to construct the triangle. 

74. To find the locus of a point equidistant from two 

converging lines without producing them till they 
meet. 

75. To draw a line that shall have a given length, be 

terminated by two circumferences and be parallel 
to the line that joins their centres. 

76. To describe a circumference that shall have the 

same center as a square and shall be described 
by the sides of the square into eight equal ares. 


40 


GEOMETRICAL EXERCISES. 


BOOK III. 

PROPORTION, SIMITAR POLYGONS. 

Define: 

1. Proportion. 

2. Extremes, Means. 

3. Third Proportional, Fourth Proportional, Mean Propor¬ 

tional. 

4. Couplet, Antecedent, Consequent. 

5 . Alternation, Inversion, Composition, Division. 

[1.] If four quantities are in proportion the product of 
the means is equal to the product of the extremes. 

Find: 

1 . A fourth proportional to A, B and C. 

2. A third proportional to A and B. 

3 . A mean proportion between A and B. 

[2.] If the product of two quantities is equal to the prod¬ 
uct of two other quantities, one pair of factors 
may be made the means and the other the ex^ 
tremes cf a proportion. 

Derive proportions from the following equations: 

1. 7 X= 11 Y. 

2. Y 2 = X Z. 

3. X 2 —Y 2 = A 2 —B 2 . 

[3.] If four quantities are in proportion, they are in pro¬ 
portion: 

1. By Alternation. 

2. By Inversion. 

3 . By Composition. 

4 . By Division. 

5. By Composition and Division. 

[4.] If four quantities are in proportion, like powers or 
like roots of these quantities are in proportion. 


GEOMETRICAL EXERCISES. 


41 


[5.] Equi-multiples or equi-submultiples of two quan¬ 
tities have the same ratio as the quantities them¬ 
selves. 

[6.] In a series of equal ratios, the sum of all the ante¬ 
cedents is to the sum of all the consequents as any 
antecedent is to its consequent. 

[7.] If all the terms of a proportion be increased or 
diminished by like parts of themselves, the results 
are in proportion. 

Define Similar Polygons. 

[8.] A line drawn parallel to one side of a triangle 
divides the other two sides proportionally. 

Case 1. When on either side, the part intercepted by the 
parallel is commensurable with that side. 

Case 2. When the part intercepted is incommensurable with 
the side. 

[9.] State and prove the conversely. 

DEMONSTRATION: 


1. The statement: 

If a straight line divides two sides of a triangle pro¬ 
portionally, it is parallel to the third side. 

2. The construction: 


A 



In the triangle A B C let the 
line D E divide the sides A D 

, ArA ,, , AD AE 
and A C so that —— =—— 

A B AC 

If D E is not parallel to B C, 
suppose another line D F to be 
parallel to B C. 





42 


GEOMETRICAL EXERCISES. 


3* 


The demonstration: 

Since D F is parallel to B C, by the previous propo- 


sition, A 5 ==A*r But by hypo- 


thesis. 


(Things that are equal to 


A B AC. 

A F_A E 
AC AC 
the same things are equal to each other.) 

Multiply both members of this equation by A C, we 
have A F= A E, which is impossible unless D F 
A D_AB . A D_AE 
AB AC A B AC 


coincide with D E, 


Q. E. D. 

[io.] If in the triangle A B C, D E is drawn parallel to the 

base B C, show that —- 5 - also that 

D B E C, 

A D_A B 

D E AC. 

[ii.] The bisector of any angle of a triangle divides the 
opposite side into parts proportional to the adja¬ 
cent sides. 

[i2.(] Mutually equiangular triangles are similar. 

[13.] Two triangles are similar when their homologous 
sides are proportional. 

[14.] Test the similarity of two triangles which have: 

1. An angle of the one equal to an angle of the other and 

the homologous including sides proportional. 

2. The three sides of the one respectively parallel to the 

three sides of the other. 

3. The three sides of the one respectively perpendicular to 

the three sides of the other. 

[15.] Test the similarity : 

1. Of two triangles which have two angles of the one equal 
„ to two angles of the other. 




GEOMETRICAL EXERCISES. 


43 


2. Of the opposite pairs of triangles formed by drawing the 

diagonals of parallelogram. 

3. Of the corresponding triangles formed by drawing the 

homologous diagonals of two similar trapeziums. 

4. Of two triangles formed by drawing perpendiculars to the 

base of a right triangle. 

5. Of the two triangles formed by drawing a perpendicular 

from the vertex of the right angle of a right triangle to 
the hypotenuse, also of either of the small triangles 
so formed to the whole triangle. 

6. Of two triangles formed by joining the extremities of two 

chords which intersect in a circle. 

7. Of two isosceles triangles which have an angle of the one 

equal to an homologous angle of the other. 

8. Of two triangles which have an angle of the one equal to 

an angle of the other and are inscribed in unequal 
circles. 

9. Of two triangles formed by drawing two secants through 

the point of tangency of two tangent circles and joining 
their extremities in each circumference. 

[16.] Join the middle points of the sides of a triangle, 
and test the similarity of each of the four triangles 
so formed with the entire triangle. 

[17.] Construct a triangle similar to a given triangle and 
having four times its area. 

[18.] Draw a circle tangent internally to another circle 
and from the point of tangency draw two chords 
of the larger circle ; join the extremities of the 
chords in each circumference and test the similar¬ 
ity of the triangles formed. 

[19.] In similar triangles, homologous altitudes have the 
same ratio as any two homologous sides. 

[20.] Two similar polygons may be divided into the same 
number of triangles similar to each other and simb 
larly placed. 


44 


GEOMETRICAL EXERCISES. 


[21.] State and prove the conversely. 

[22.] The perimeters of two similar polygons have the 
same ratio as any two homologous sides. 

[23.] If a perpendicular is drawn from the vertex of the 
right angle of a right triangle to the hypotenuse: 

1. The perpendicular is a mean proportional between the 

segments of the hypotenuse. 

2. Either side about the right angle is a mean proportional 

between the whole hypotenuse and the adjacent segment. 

24. Divide the diameter of a circle in the ratio of 2 to 3 
and find a mean proportional between the 
segments. 

T25.] If through a fixed point in a circle a chord is drawn, 
the product of its segments is a constant in what¬ 
ever direction the chord be drawn. 

[26.] If from a point without a circle a secant be drawn, 
the product of the whole line by its external seg¬ 
ment is a constant in whatever direction the secant 
be drawn. 

[27.] If from a point without a circle a tangent and a 
secant be drawn, the tangent is a mean propor¬ 
tional between the whole secant and its external 
segment. 

28. In any triangle the square of the bisector of an 

angle is equal to the product of the including 
sides diminished by the product of the segments 
into which the bisector divides the third side. 

29. In any triangle the product of two sides is equal to 

the product of the altitude upon the third side, by 
the diameter of the circumscribed circle. 

30. Divide a given straight line in the proportion of 1, 

3 and 5. 


GEOMETRICAL EXERCISES. 


45 


31. Find the fourth proportional to three lines in the 

proportion 1, 3 and 5. 

32. Find a third proportional to two lines in the pro¬ 

portional of 2 to 3. 

33. Find a mean proportional between two lines in the 

proportion of 2 to 3. 

[34-] To find : 

1. A fourth proportional to three given lines. 

2. A third proportional to two given lines. 

3. A mean proportional between two given lines. 

35, Designate three straight lines by A B and C, making 

A>B and B>C, and construct: 

i- ^ 3- V AB. 5 - A+i/ABT 7. j/A(B—C.) 

2 ' X 4- 6. A—|/ABX8. j/A'—B 2 . 

36. Designate a straight line as one unit and construct 

1. 4 / 2 . 3. 1 / 6 . 5■ 

2. 1 / 5 . 4. 6. 1 / 3 + 1 / 3 . 

7 - 

[37.] To divide a given straight line in extreme and mean 
ratio. 

[38.] Upon a given straight line to construct a polygon 
similar to a given polygon. 

39. To inscribe in a given circle: 

1. A triangle similar to a given triangle. 

2. A quadrilateral similar to a given quadrilateral. 

40. A B is the upper base of the trapezoid A B C D and 

E is the point of intersection of its diagonals. 
Show that the triangles A E B and C E D are 
similar. 





46 


geometrical exercises. 


41. If two straight lines are cut by any number of parallel 

lines, the corresponding segments are proportional. 

42. An equilateral triangle is circumscribed about a cir¬ 

cle whose radus is B. Find the distance from the 
center of the circle to the vertex of the triangle. 

43. To inscribe in a given circle a trapezium similar to a 

given trapezium. 

44. The common chords of three mutually intersecting 

circles meet in a point. Is there any exception to 
this principle ? 


geometrical exercises. 


47 


BOOK IV. 

COMPARISON AND MEASUREMENT OF SUR¬ 
FACES. 

1. Define Equivalent Figures. 

2. Discriminate between Equal Figures and Equivalent 

Figures. 

[i.] Rectangles having equal altitudes are to each other 
as their bases. 

1. The areas of two rectangles of equal altitudes are 

respectively 20 and 21. The base of the larger is 
11. Find the base of the smaller. 

2. Two rectangles with equal bases are to each other as 

their altitudes. 

[2.] Any two rectangles are to each other as the products 
of their bases and altitudes. 

[3.] The area of any rectangle is equal to the product of 
its base and altitude. 

[4.] The area of any parallelogram is equal to the prod¬ 
uct of its base and altitude. 

[5.] The area of a triangle is equal to one-half the prod¬ 
uct of its base and altitude. 

1. Construct a parallelogram which shall have 
double the area of a given triangle. 

2; Construct a figure which shall be composed of 
a scalene triangle and three (not equal) 
parallelograms, each twice the area of the given 
triangle. 


48 


GEOMETRICAL EXERCISES. 


3. What relation exists between two triangles: 

1. Having equal bases or altitudes? 

2. Having equal bases and altitudes? 

3. Having their vertices in the same point and their bases in 

the same straight line? 

4. Divide a triangle into two equal parts by 

means of a line from the vertex to the base ; into 
three equal parts by means of lines from the ver¬ 
tex to the base ; into n equal parts. 

5. Test the equivalence: 

1. Of the four triangles formed by the diagonals of a rect¬ 

angle. 

2. Of the four triangles formed by the diagonals of a par¬ 

allelogram. 

3. Of the six triangles formed by the medians of a triangle. 

[6.] The area of a trapezoid is equal to half the product 

of the sum of the parallel sides by its altitude. 

[7.] Formulate the area of a circumscribed polygon in 
the terms of P (its perimeter) and R (the radius 
of the inscribed circle.) 

1. The side of an equilateral triangle is m and the radius of 

the inscribed circle is 6, find the area of the triangle. 

2. The area of a polygon is 64 and its perimeter is 32. Find 

the radius of the inscribed circle. 

[8.] Two triangles which have an angle of the one equal 
to an angle of the other are to each other as the 
product of the including sides. 

[9.] Similar triangles are to each other as the squares of 
their homologous parts. 

1. Similar polygons are to each other as the squares of their 

homologous sides. 

2. The homologous sides of similar polygons are to each 

other as the sqnare roots of the area of the polygons. 

[10.] The square described on the hypotenuse of aright 


GEOMETRICAL, EXERCISES. 


49 


triangle is equal to the sum of the squares 
described on the other two sides. 

In the following exercises H designates the hypote¬ 
nuse, B the base, A the altitude, P the perimeter 
and S the area. 

1. Express H in terms of A and B. 

2. Express B in terms of A and H. 

Given: 

3. B, X; A, 2 X; to find H. 

4. H, 10 X; B, 6 X, to find A. 

5. B, X+Y; A, (X—Y) to find H. 

6. H, X 2 +Y 2 ; A, (X 2 —Y 2 ), to find B. 


7 . H, 


M , N. -o M N 

Is M’ ’ N M’ 


to find A. 


8. A, 27; B, 36, to find H. 

9. (A-f-B), 49; H, 35; to find A and B. 

10. (H-f-A), 24; B, 12, to find IT and A. 

11. (H—B), 6; A, 18, to find H and B. 

12. P, 48; S 96, to find the sides. 

13. H, 40; S, 384, to find the sides. 

14. A tree 90 feet high is broken off, the trunk remaining on 

the stump. The top strikes the ground 30 feet from the 
stump ; find the height of the stump. 

15. Find the ratio : 

1. Of the diagonal to the side of a square. 

2. Of the side of an equilateral triangle to its altitude. 

16. The diagonal of a square is 20 ; find its area. 

17. The side of an equilateral triangle is 6 ; find its area. 

18. The area of an equilateral triangle is 100 ; find its altitude. 
IQ. Show that any triangle whose sides are in the proportion 

of 3, 4 and 5 is a right triangle. • 

20. Find the length of a chord which subtends a quadrant in a 
circle whose radius is 10. 


50 


geometrical exercises. 


21. A and B are the sides of an obtuse triangle. A perpendic¬ 
ular is drawn from the obtuse angle to the base, dividing 
it into two parts, C and D. Show : 

1. That A 2 +D 2 =B 2 +C 2 . 

2. That A+B: C+D :: C—D: A—B. 

Define and illustrate: 

1. The Projection of a point upon a line. 

2. The Projection of a line upon another line. 

[n.] In any triangle the square of a side opposite an 

acute angle is equal to the sum of the squares of 
the other two sides diminished by twice the prod¬ 
uct of one of those sides and the projection of the 
other upon that side. 

[i2.] State and demonstrate a similar proposition for the 
value of the square described on the side opposite 
the obtuse angle of an obtuse triangle. 

13. Represent two straight lines by X and Y and show 

geometrically: 

1. That (X+Y) 2 =X 3 +2 X Y+Y 2 . 

2. That (X—Y) 2 =X 2 —2 X Y+Y 2 . 

3. That (X+Y) (X—Y)—{X 2 —Y 2 .) 

14. Represent three straight lines by A B and C, making 

A>B, and B>C, and construct 

1. A 2 +B 2 . 3. A 2 +B 2 +C 2 . 

2. A 2 —B 2 . 4. A 2 +B 2 —C 2 . 

[15.] To construct a square equivalent: 

1 . To a given parallelogram. 

2 . To a given triangle. 

3 . To four times a given square. 

4 . To four times a given triangle. 

[16.] To construct a triangle equivalent to a given 
polygon. 


GEOMETRICAL EXERCISES 


51 


17. To construct a square equivalent to a given 

pentagon. 

18. To construct a triangle equivalent: 

1 . To a given square. 

2 . To a given octagon. 

3 . To a given quadrilateral, one of whose angles is re¬ 

entrant. 

4 . To a given polygon having the form of a five-pointed 

star. 

[19.] To construct a polygon similar to two given similar 
polygons and equivalent to their sum. 

[20.] To construct a polygon similar to two given similar 
polygons and equivalent to their difference. 

21. To construct a square that shall be two-thirds of a 

given square. 

22. To construct a polygon that shall be similar to a 

given polygon and shall have two-thirds its area. 

23. To construct a parallelogram equivalent to a given 

square and having the sum of its base and altitude 
equivalent to a given line. 

24. To construct a parallelogram equivalent to a given 

square and having the difference of its base and 
altitude equivalent to a given line. 

25. To construct a square equivalent to the sum of two 

given unequal scalene triangles. 

26. On the sides B D and C D of the rectangle ABC 

D, the squares BBFD and D G H C are respect¬ 
ively drawn, show that the diagonals E D and D 
H form a straight line. If the adjacent sides of 
the rectangle are 4 and 3, find the length of 
E H. 

27. The square on the altitude of an equilateral triangle 

is three-fourths of the square on the side. 


52 


geometrical EXERCISES. 


[28.] If from the vertex of a triangle a median be drawn to 
the base : 

1 . The sum of the squares of the two sides is equal to twice 

the square of half the base, increased by twice the 
square of the median. 

2 . The difference of the squares of the sides is equal to twice 

the product of the base by the projection of the median 
upon the base. 

29. The sum of the squares on the sides of a parallelo¬ 

gram is equal to the sum of the squares on the 
diagonals. 

30. The sum of the squares on the sides of any quadri¬ 

lateral is equal to the sum of the squares of the 
diagonals, increased by four times the square of 
the line joining the middle points of the 
diagonals. 

31. The sum of two opposite triangles formed by joining 

any point in a parallelogram with its vertices is 
equal to half the parallelogram. 

32. To construct a polygon similar to a given polygon 

P, and equivalent to a given polygon O. 

33. Given the area of a right triangle, and an acute 

angle, to construct the triangle. 

34. Given a scalene triangle to construct a triangle that 

shall be similar to it and have double its area. 

35. Given a scalene triangle to construct an equilateral 

triangle of the same area. 

36. To bisect a given triangle by means of a line per 

pendicular to the base. 


GEOMETRICAL EXERCISES. 


53 



It is required to bisect the triangle A B C by means of a 
line perpendicular to the base B C. From the ver¬ 
tex draw the median line A F and the perpendic¬ 
ular A D. Find a mean proportional between 
C D and C F, and lay off its length C K on C B. 
Draw the perpendicular G E. It divides the tri¬ 
angle into two equal parts. 

THE DEMONSTRATION. 

1. A B A F=A A C F, the triangles 

having equal bases and equal altitudes. 

2. A A C D: A A C F:: C D: C F. 

Triangles having equal altitudes are to 
each other as their bases. 

3. A A C D: A G C E:: CD*: CE. Sim¬ 

ilar triangle are to each other as the 
squares of their homologous sides. 

4. A A C D: A G C E:: CF: C DXC F, 

since C E is a mean proportional between 
C D and C F. 

5. A ACD: AG-CE:: CD: CF, dividing 

both terms of second couplet by C D. 

In proportions (2) and (5) the corresponding terms are 
identical except the second term of the first couplet 
• •• A G C E=A A C F =>4 A A B C. Q. E. F. 






54 


geometrical exercises. 


37. To cut off one-third of a triangle by means of a 

line perpendicular to the base. 

38. To divide a triangle in the proportion of M to N by 

means of a line perpendicular to the base. 

39. To bisect a given triangle by means of a line par¬ 

allel to one of its sides. 

40. To trisect a given triangle by means of lines drawn 

parallel to one of its sides. 

41. The square described on any straight line is equal 

to n times the square described on — part of that 

n 

line. 

4 2. The area of any rectangle is equal to half the prod¬ 

uct of the diagonals of the squares described on 
two adjacent sides. 

43. Three times the sum of the square on the sides of a 

triangle is equal to four times the sum of the 
squares on the medians. 

44. The equilateral triangle described on the hypot¬ 

enuse of a right triangle is equal to the sum of 
the equilateral triangles described on the other two 
sides. 

45. State and demonstrate a like principle for three sim¬ 

ilar polygons of n sides, described upon the re¬ 
spective sides of a right triangle. 


geometrical exercises. 


55 


BOOK V. 

REGULAR POLYGONS. 

THE AREA OF THE CIRCLE- 
Define Regular Polygons. 

[i.] If the circumference of a circle be divided into 71 
equal parts: 

1. The chords joining the successive points of division will 

form a regular inscribed polygon of M sides. 

2 . The tangents drawn through the successive points of 

division will form a regular circumscribed polygon of M 
sides. 

[2.] Given the side of a regular inscribed polygon of M 
sides: 

1 . To fiud the side of a regular inscribed polygon of 2 M 

sides. 

2 . To find the side of a regular circumscribed polygon of 

2 M sides. 

[3.] Given the side of a regular inscribed polygon of 2 M 
sides: 

1 . To find the .side of a regular inscribed polygon of M 

sides. 

2 . To find the side of a regular circumscribed polygon of N 

sides. (Two methods.) 

[4.] Can a circle: 

1 . Be circumscribed about any regular polygon? 

2 . Be inscribed in any regular polygon? 

1. Define: 

1 . The Center of a Regular Polygon. 

2 . The Radius of a Regular Polygon. 

3 . The Apothem of a Regular Polygon. 

2. What is the measure of the central angle: 

1 . Of an inscribed equilateral triangle? 


56 


geometrical exercises. 


2 . Of a regular inscribed pentagon? 

3 . Of a regular inscribed heptagon? 

4. Of a regular inscribed polygon of N sides? 

3. What is the measure of one of the equal angles of a 

polygon of 5 sides? Of 7 sides? Of N sides? 

4. If B is the central angle of a regular inscribed poly¬ 

gon of M sides, and A is one of the equal angles 
of the polygon, show that B is the supplement 
of A. 

[5.] Regular polygons of the same number of sides are 
similar. 

[6.] The perimeters of two regular polygons are to each 
each other: 

1 . As the radii of their circumscribed circles. 

2 . As the radii of their inscribed circles. 

[7.] The areas of two regular polygons are to each 
other: 

1 . As the squares of the radii of their circumscribed circles. 

2 . As the squares of the radii of their inscribed circles. 

[8.] If the number of sides of a regular circumscribed 

polygon be indefinitely increased, its perimeter is a 
decreasing variable, and the circumference of the 
circle is its limit. 

[9.] State and prove a like proposition concerning: 

1 . The area of the circumscribed polygon. 

2 . The perimeter of the inscribed polygon. 

3 . The area of the inscribed polygon. 

[10.] Two circumferences have the same ratio as their 
radii. 

1 . The ratio of the circumference to the diameter of a circle 

is a constant. 

2 . Derive a formula for C (the circumference of a circle) in 

terms of R (its radius). 


geometrical exercises. 


57 


[ii-] The area of a regular polygon is equal to one-half 
the product of its apothem by its perimeter. 

[i2.] The area of a circle is equal to one-half the product 
of its radius by its circumference. 

1. Derive a formula for A (the area) of a circle in terms of R. 

2. Two circles are to each other as the squares of their 

radii. 

[13.] Of two similar sectors: 

1. The arcs are to each other as the radii. 

2. The areas are to each other as the squares of the radii. 
[14.] To inscribe in a given circle an equilateral triangle. 
[15.] To inscribe in a given circle a square. 

[16.] To inscribe in a given circle a regular hexagon. 

In the circle whose center is O, inscribe the regular hexa¬ 
gon A B C D E F. 

1. Divide the hexagon into two equal isosceles tropezoids. 

2. Divide it into three equal rhombuses. 

3. Join A E and B D. What form of quadrilateral is A B D E? 

4. Draw F O and O C. Is F O C a straight line ? 

5. If F O C cut A E and B D at I and J, are the figures A B 

I J and IJ F D squares ? 

6. At what angles do the diagonals A D and B E intersect ? 

7. Draw the diagonals A C, C E and A E. What form of 

triangle is A C E ? 

8. Draw the diagonals B D, D F and B F. Is the hexagon 

MNPQRS formed by the intersections of the diagon¬ 
als regular ? D. 

9. Is F B trisected at the points M and N ? D. 

10. Test the equality and equivalence of the triangles F A M, 

MAN, and A N B. 

11. Find the ratio of the regular circumscribed to the regular 

inscribed hexagon. 

12. About the circle describe six circles each equal to it, tan. 

gent externally to it, and each tangent to two of the tan¬ 
gent circles, join the centers forming the hexagon M N 


58 


GEOMETRICAL, EXERCISES. 


PQRS, is it a regular hexagon? Find its ratio to the 
regular inscribed hexagon. 

13. Draw common tangents to each pair of outer circles. 

Is the figure formed a regular hexagon ? 

[17.] To inscribe a regular decagon in a given circle. 

1. To find the side of a regular pentagon. 

2. To find the side of a regular polygon of twenty sides. 
[18.] To inscribe a regular pentadecagon in a given circle. 

How many and what regular polygons of fewer 
than fifty sides can be inscribed in a circle by geo¬ 
metrical processes ? 

[19.] Given a regular polygon, to inscribe in a given 
circle a regular potygon of the same number of 
sides. 

[20.] Derive an expression for the chord of half an arc in 
terms of the chord of the whole arc and the radius 
of the circle. 

[21.] To find the numerical value of r, (the ratio of the 
circumference ot the circle to its diameter.) 

Are all equiangular polygons regular? Are all equi¬ 
lateral polygons regular? 

22. Construct, if possible, an equilateral polygon: 

1. That cannot be inscribed in a circle. 

2. That cannot be circumscribed about a circle. 

23. Construct, if possible, an equiangular polygon: 

1. That cannot be inscribed in a circle. 

2. That cannot be circumscribed about a circle. 

24. Find in terms of the radius of the circumscribed 

circle, the area: 

1. Of an equilateral triangle. 

2. Of a square. 

3. Of a regular hexagon. 

4. Of a regular octagon. 

5. Of a regular decagon. 


GEOMETRICAL EXERCISES. 


59 


25. Find the ratio of the circumscribed regular hexa¬ 

gon to the inscribed regular hexagon. 

26. Find the ratio of the area of a regular dodecagon to 

the square on the radius of the circumscribing cir- 
cle. 

27. Inscribe a regular hexagon in a given equilateral 

triangle. 

28. The circumference of a circle is 20; find its area. 

29. The area of a circle is 6 2 8 3. 2; find its circumfer¬ 

ence. 

30. To find the largest wheel that can be cut from a 

board in the form of an isosceles tropezoid: 

1. When the altitude is less than the mean base. 

2. When the altitude is greater than the mean base. 

31. The perimeter of a triangle is 50, and its area is 

100; find the area of the inscribed circle. 

32. Given an equilateral triangle, to find the ratio of 

the circumscribed to the inscribed circle. 

33. Given the side of a regular pentagon to construct 

the pentagon. 

34. If from any point within a regular polygon of M 

sides, perpendiculars be drawn to the sides, their 
sum is equal to M times the radius of the 
inscribed circle. 

35. In the inscribed hexagon A B C D B F, prove >A 

+>C+>E=>B-f>D+>F. 

36. The radius of a circle is 20; find the area of a sec¬ 

tor of 75 0 of the circle. 

37. Given three equal mutually tangent circles whose 

radius is 10; to find the area of the space without 
the circles but included by the circumferences. 

38. The space included (as in 38) by three equal mutu- 


60 


geometrical exercises. 


ally tangent circles is ioo square feet. Finu 
length of the radius. 

39. By means of concentric circumferences to divide a 

given circle: 

1. Into two equivalent parts. 

2. Into three equivalent parts. 

3. Into four equivalent parts, 

4. Into N equivalent parts. 

40. To inscribe three equal mutually tangent circles in 

a given equilateral triangle. 

41. To inscribe three mutually tangent circles in an 

triangle. 

42. To inscribe three equal mutual^ tangent circles in 

a given circle. 

43. To inscribe four equal circles in a given square: 

1. So that each circumference shall touch two circumfer¬ 

ences and one side of the square. 

2. So that each circumference shall touch two circumfer¬ 

ences and two sides of the square. 

44. To inscribe four equal circles in a given circle so 

that each circumference may touch two of the 
equal circumferences. 

45. To inscribe six equal circles in a given equilateral 

triangle. 

46. To describe about a given circle: 

1. Three equal circles. 

2. Four equal circles. 

3. N equal circles. 

47. With given radii, to describe three mutually tan¬ 

gent circles. 

48. To inscribe a circle in the space bounded by the 

intercepted arcs of fl three equal mutually tangent 
circles. 


geometrical exercises. 


61 


BOOK VI. 

PLANES. 

1. Define a Plane. 

2. How is a plane generated? 

3. Define the Intersection of two planes. 

4. When is a straight line perpendicular to a plane? 
Parallel to a plane? 

5. When are two planes perpendicular to each ot'ier? 
Parallel to each other? 

6. How is a plane determined? Four methods. 

7. Can a plane be passed through any three points in 

space? Can a plane be passed through any two • 
lines in space? 

8. Define the Locus of a Point in space. 

9. What is the locus of a point in space equidistant: 

1. From two given points? 

2. From two given straight lines which lie in the same plane? 

3. From two given parallel planes? 

4. From two given intersecting planes? 

10. Define the Projection of a Point on a Plane. The 

Projection of a Line on a Plane. 

11. Define Mathematical Symmetry. 

12. Define and illustrate by diagram : 

(See Vocabulary.) 

1. Two points symmetrical with reference: 

1. To a point. 

2. To an axis. 

3. To a plane. 

2. Two lines symmetrical with reference: 

1. To a point. 

2. To an axis. 

3. To a plane. 


62 


GEOMETRICAL EXERCISES. 


3. Two planes symmetrical with reference: 

1. To a point. 

2. To an axis. 

3. To a plane. 

13^ What are Symmetrical Figures? 

14. Classify the capital printed letters of the alphabet 
as symmetrical with reference to a point, symmet¬ 
rical with reference to an axis, and not symmet¬ 
rical. 

Let the test for each letter be whether it can be written as 

a symmetrical figure or not. 

[1.] From a given point without a plane only one perpen¬ 
dicular can be drawn to the plane; and from a 
given point in a plane only one perpendicular can 
be erected to the plane. 

[2.] Two straight lines perpendicular to the same plane 
are parallel. 

[3.] If from a point without a plane, a perpendicular and 
oblique lines be drawn to the plane: 

1. The perpendicular is the shortest line to the plane. 

2. Oblique lines meeting the plane at equal distances 

from the foot of the perpendicular are equal. 

3. Of two oblique lines meeting the plane at unequal 
distances from the foot of the perpendicular, the 
one which meets it at the greater distance is the 
longer. 

[4.] A line perpendicular to two lines of a plane at their 
point of intersection is perpendicular to the 
plane. 

[5.] If through the foot of a perpendicular to a plane a 
line be drawn intersecting any line of the plane at 


GEOMETRICAL EXERCISES. 


63 


right angles and the point of intersection be 
joined with any point of the perpendicular, the 
last line will be perpendicular to the line of the 
plane. 

What is a Dihedral Angle? 

Define the Face of a dihedral angle; the Edge of a 
dihedral angle. 

What is the Plane Angle of a dihedral angle ? 

Define a Secant Plane. 

[6.] The intersection of two planes is a straight line. 

[7.] The intersections of parallel planes by a secant 
plane are parallel. 

[8.] What relation subsists between two vertical dihedral 
angles? 

If two parallel planes are cut by secant planes, what re¬ 
lation subsists between: 

1. Alternate interior dihedral angles ? 

2. Exterior interior dihedral angles ? 

3. Interior dihedral angles on the same side of the secant 

plane ? 

State the conversely of 1, 2 and 3. 

What relation subsists: 

1. Between two dihedral angles whose faces are respectively 

parallel each to each. 

2. Between two dihedral angles whose faces are respectively 

perpendicular each to each. 

[9.] If two planes are perpendicular to the same straight 
line they are parallel. 

[10.] Parallel lines included between parallel planes are 
equal. 

[11.] Two straight lines cut by three parallel planes are 
divided proportionally. 

[12.] If two intersecting planes are each perpendicular to 


64 


GEOMETRICAL EXERCISES. 


a third plane, their intersection is also perpendic¬ 
ular to the third plane. 

[13.] If two angles not in the same plane have their sides 
respectively parallel and lying in the same direc¬ 
tion, the angles are equal and their planes are 
parallel. 

[14.] Every point in the bisecting plane of a dihedral 
angle is equally distant from the sides of the 
angle. 

[15.] What is the locus of a point in space at a given dis¬ 
tance from a given plane ? 

16. To find the height of the ceiling of a room by 
means of a pole longer than the perpendicular dis¬ 
tance from floor to ceiling. 

Define a Trihedral Angle, a Polyhedral Angle. 

Define the terms Edge, Face, Vertex, Face Angle of 
a Polyhedral Angle. 

Define Rectangular, Bi-rectangular, Tri-rectangular, 
Equi-angular, Isosceles and Scalene trihedral 
angles. 

[17.] The sum of any two face angles of a trihedral angle 
is greater than the third angle. 

[18.] The sum of all the face angles of a polyhedral angle 
is less than four right angles. 

[19.] Two trihedral angles are either equal or symmetri¬ 
cal and equivalent, if the three face angles of the 
one are equal respectively to the three face angles 
of the other. 

20. The three bisecting planes of a trihedral angle meet 
in the same straight line. 


GEOMETRICAL EXERCISES. 65 

21. Find the locus of a point equidistant from the edges 

of a trihedral angle. 

22. To find the locus of a point equidistant from two 

given points and also equidistant from two given 
intersecting planes. 


GEOMETRICAL EXERCISES. 


66 

BOOK VII. 

SOLIDS. 

Define: 

1. A Polygon. 

2. The Face, Edge, Vertex of a Polyhedron. 

3. The Diagonal of a Polyhedron. 

4. A Tetrahedron. What is a Polyhedron bounded 

by five faces called ? by six faces? by seven? 
by 8? by 9? by 10? by 11? by 12? by 20 faces? 
See table, Page 84. 

5. A Prism. 

6. The Base, Lateral Face, Lateral Edge, Basal 

Edge of a Prism. 

7. The Lateral area of a Prism. 

8. An Oblique Prism. A Right Prism. A Regular 

Prism. 

9. A Cylindrical Surface. 

10. Generatrix, Directrix, Element. 

11. A Cylinder. 

12. A Cylinder of Revolution. Define: 

1. By its generation. 

2 . As a prism. 

13. A right section of a cylinder. 

14. The altitude of a cylinder. 

[1.] The sections of a prism made by parallel planes are 
equal polygons. 

[2.] Derive a formula for the lateral area of a prism in 
terms of E (an element of the prism) and P (the 
perimeter of a right section.) 

[3.] Every section of a prism made by a plane embrac¬ 
ing an element is a parallelogram. 


GEOMETRICAL EXERCISES. 


67 


[4.] Any two parallel sections of a cylinder are equal. 

[5.] Derive a formula for the lateral area of a cylinder of 
revolution in terms of E (an element of the cylin¬ 
der) and R (the radius of a right section). 

Define: 

1. A Pyramid. A Regular Pyramid. 

2. The Base, Lateral Face, Lateral Edge, Basal 

Edge, and Vertex of a pyramid. 

3. The Slant Height of a pyramid. 

4. A Frustum of a pyramid. 

5. The Lateral face of a frustum. 

6. Lower Base, Upper Base, Lateral Surface, Slant 

Height of a Frustum. 

7. A Conical surface. 

8. Generatrix, Directrix, Vertex. 

9. A Circular Cone; a Cone of Revolution. 

10. A Pyramid inscribed in a cone. 

11. The Slant Height of a cone. 

12. A Frustum of a cone. 

[6.] Derive a formula for the lateral area of a regular 
pyramid in terms of P (the perimeter of the base) 
and H (the slant height). 

[7.] Derive a formula for the lateral area of a frustum of a 
regular pyramid in terms of P (the perimeter of 
the lower base) p (the perimeter of the upper base, 
and H (the slant height). 

[8.] Every section of a cone made by a plane embracing 
one of its elements is a triangle. 

[9.] Every section of a cone of revolution made by a plane 
passed parallel to its base is a circle. 

[10]. Derive a formula for the lateral area of a cone of 


68 


GEOMETRICAL, EXERCISES. 


revolution in terms of R (the radius of the base) 
and H (the slant height). 

[11] . The surfaces, lateral or entire, of two similar cones 

of revolution are to each other as the squares of 
their altitude or as the square of the radii of their 

bases. 

[12] Derive a formula for the lateral area of a frustum of 

a cone of revolution in terms of R (the radius of 
the lower base) r (the radius of the upper base) 
and H (the slant height of the frustum). 

Define: 

1. The Altitude of a prism, cylinder, or frustum. 
The Altitude of a pyramid or cone. 

2. A Truncated Prism. 

3. A Triangular Prism. How is a prism, the base 

of which has four sides, designated ? 5 sides ? 
6? 7? 8? 9? 10? 11? 12? 20 sides? 

4. A Parallelopiped. An Oblique Parallelopiped. 

A Right Parallelopiped. A Rectangular Par¬ 
allelopiped. A Cube. 

5. How are pyramids designated with reference to 

the number of sides which form their bases. 
[13.] If two prisms have three faces, including a trihedral 
angle of the one respectively equal to three faces 
.including a trihedral angle of the other and simil¬ 
arly placed, the prisms are equal: 

1. Test the equality of two truncated prisms in which the 

above conditions are fullfilled. 

2. Test the equality of two prisms that have equal bases and 

altitudes. 

[14.] An oblique prism is equivalent to a right prism 
whose base is a right section of the oblique prism 


GEOMETRICAL EXERCISES. 


69 


and whose altitude is equal to a lateral edge of the 
oblique prism. 

[15.] The opposite faces of a parallelopiped are equal and 
parallel. 

[16.] The plane embracing two diagonally opposite edges 
of a parallelopiped divides it into two equivalent 
triangular prisms. 

[17.] Two rectangular parallelopipeds having equivalent 
bases are to each other as their altitudes: 

Case 1. When the altitudes are commensurable. 

Case 2. When the altitudes are incommensurable. 

[18.] Two rectangular parallelopipeds having equal alti¬ 
tudes are to each other as their bases: 

1. Two rectangular parallelopipeds having two dimensions 

in common are to each other as their third dimensions. 

2. Two rectangular parallelopipeds having one dimension in 

common are to each other as the products of the other 
two dimensions. 

[19.] Two rectangular parallelopipeds are to each other 
as the products of their three dimensions. 

[20 ] The volume of a rectangular parallelopiped is equal 
to the product of its three dimensions. 

1. The surface of a cube is 12, find its volume. 

2. The volume of a cube is 27, find its surface. 

3 What is the ratio of the volume to the surface of a cube 
whose edge is E ? 

[21.] The volume of any parallelopiped is equal to the 
product of its base by its altitude. 

[22.] The volume of a triangular prism is equal to the 
product of its base by its altitude. 

[23.] The volume of any prism is equal to the product of 
its base by its altitude. 

1. Two prisms are to each other as the products of their 
bases and altitudes. 


0 


GEOMETRICAL EXERCISES. 


2. Prisms having equivalent bases are to each other as their 

altitudes. 

3. Prisms having equivalent altitudes are to each other as 

their bases. 

4. Two prisms are equivalent if they have equivalent bases 

and equal altitudes. 

[24.] The volume of a cylinder is equal to the product of 
its base by its altitude. 

[25.] Derive a formula: 

1. For the volume of a prism in terms of B (the base) and 

A (the altitude). 

2. For the volume of a cylinder of revolution in terms of R 

(the radius of the base) and A (the altitude). 

26. Each edge of a right triangular prism is 6, find its 

volume. 

27. The volume of a regular right triangular prism is 

100, its altitude is 10, find its basal edge. 

28. Each side of the base of a regular hexagonal prism 

is 9, and its altitude is 8, find its volume. 

29. Represent the dimensions of a rectangular parallel- 

opiped by E. H and B, and derive a formula for its 
diagonal. 

1. The dimensions of a room 30, 20 and 120ft. respectively, 

find the length of its diagonal. 

2. Find the diagonal of a cube whose edge is 12. 

30. Find the shortest route by which a fly may walk 

from a lower corner of a room to the diagonally 
opposite upper corner. 

31. Find the volume of a cylinder the radius of whose 

base is 12 and whose altitude is 7. 

32. The diameter of a cylindrical cistern is 5ft. and its 

depth is 20ft., how many gallons will it contain? 

33. The length of a right cylinder is equal to the diam- 


geometrical exercises. 


71 


eter of its base. If its surface is ioo, what is its 
volume? 

34. Find the radius of the base of a right cylinder 
whose altitude is twice the diameter of the base, 
and which has as many square inches in its entire 
surface as it has cubic inches in its volume. 

[35.] If a pyramid is cut by a plane parallel to its base. 

1. The edges and altitude are cut proportionally. 

2. The section is a polygon similar to the base. 

1 . If a pyramid is cut by a plane parallel to the base, the 

area of the section is to the area of the base as the 
square of the distance from the vertex to the section 
is to the square of the altitude of the pyramid. 

2 . If two pyramids have equal altitudes and equivalent 

bases, sections made by planes parallel to their bases 
and at equal distances from their vertices are equiv¬ 
alent. 

[36.] Two triangular pyramids having equivalent bases 
and equal altitudes are equivalent. 

[37.] Find the ratio of a triangular prism to a triangular 
pyramid having the same base and altitude. 

1. The basal edges of a triangular prism are 5, 5 and 8, its al¬ 

titude is 12, find its volume. 

2. Derive a formula /or nnding the volume of anypyramid in 

terms of B (its base) and A (its altitude). 

3. The volumes of two pyramids are to each other as the pro¬ 

ducts of their bases and altitudes. 

4. Pyramids having equal altitudes are to each other as their 

bases, and pyramids having equivalent bases are to each 
other as their altitudes. 

5. Pyramids having equivalent bases and equal altitudes are 
• equivalent. 

[38.] Derive a rule for finding the volume of a cone in 
terms of its base and altitude. 


72 


geometrical exercises. 


1. The radius of the base of a cone is 12, and its altitude is 12 

find its volume. 

2. Derive a formula for finding the volume of a cone of revo¬ 
lution in terms of R (the radius of the base) and A 
(the altitude). 

[39-] The frustum of a triangular pyramid is equivalent to 
the sum of three pyramids whose common altitude 
is the altitude of the frustum, and whose bases are 
the lower base of the frustum, its upper base, and a 
mean proportional between the bases. 

[40.] State and demonstrate a like theorem for the volume 
of any pyramid. 

1. Each edge of the lower base of a frustum of a square pyra¬ 

mid is 9; each edge of the upper base is 7, and its altitude 
is 20; find its volume. 

2 . Derive a formula for finding the volume of a frustum of a 

right pyramid in terms of B (the lower base) b (the up¬ 
per base) and H (the altitude of the frustum). 

[41.] Derive a formula for finding the volume of a frustum 
of a cone of revolution in terms of R (the radius 
of the lower base) r (the radius of the upper base) 
and H (the altitude of the frustum). 

[42.] A circular tank is 7 ft. in diameter at the bottom, 4 
ft. at the top and 6 ft. deep. How many gallons 
will it contain? 

[43.] The lateral surface of a cone is 78.54. The radius 
of the base is 8, find the volume of the cone. 

Define a Regular Polyhedron. 

[44.] Show that five and only five regular polyhedrons 
may be constructed. 

[45.] To construct: 

1 . A regular tetrahedron. 

2. A regular hexahedron. 



geometrical exercises. 


73 


3 . A regular octahedron, 

4. A regular dodecahedron. 

5. A regular icosahedron. 


[46.J Construct from card-board models of the five regu¬ 
lar polyhedrons. 

Cut the card-board as shown in the figures below : 





sum of three pyramids which have for a common 
base the base of the prism and for vertices 
the vertices of the inclined section. 

Define Similar Polyhedrons. 

[48.] Test the similarity of two regular polyhedrons hav¬ 
ing the same number of faces. 

[49.] Two similar polyhedrons maybe divided into the 
same number of tetrahedrons, similar to each 
other and similarly placed. 

[50. ] The volumes of two tetrahedrons which have a tri¬ 
hedral angle of the one equal to a trihedral angle 
of the other are to each other as the products of 


















74 


geometrical exercises. 


the respective edges of the equal trihedral angles. 
[51.] The volumes of two similar tetrahedrons are to each 
other as the cubes of their homologous edges. 

[52. ] The volumes of two similar polyhedrons are to each 
other as the cubes of their homologous edges. 

[53.] Similar cylinders, cones, or frustums of cones are 
to each other as the cubes of their homologous 
altitudes or of their homologous radii. 

54. Find the volume: 

1 . Of a regular tetrahedron whose edge is 2 . 

2 . Of a regular octahedron whose edge is 10 . 

55. The diameter of the base of a cone is 20, the 

altitude 16. A plane is passed parallel to the 
base dividing the cone into two equivalent parts. 
Find the altitude of each part. 

56. The edge of a regular polyhedron is 5. Find the 

edge of a similar polyhedron one tenth as large. 
57.. The volume of a regular octahedron is 300. Find 
its edge. 

58. The dimensions of a rectangular patallelopiped are 

respectively 4, 5 and 6. Find the dimensions of a 
similar solid eight times as large. 

59. Two similar cylinders are to each other as M to N. 

The diameter of the first is A, and its altitude is 
B. Find the volume of the second. 

60. To divide a given tetrahedron into four equivalent 

tetrahedrons. 

61. The sum of the squares of the four diagonals of a 

parallelopiped is equivalent to the sum of the 
square of the twelve edges. 

62. If the altitude of a right cylinder is equivalent to 

the diatne ter of the base, the volume is equivalent 


GEOMETRICAL EXERCISES. 


75 


to the total area multiplied by one-third of the 
radius. 

63 Each edge of a triangular prism is equal to the 
edge of a cube. Find the ratio of the volume of 
the cube to the volume of the prism. 

64. Find the volume of a regular hexagonal prism, its 

basal edge 10, and its lateral edge 16. 

65. Find the volume of a square pyramid, each of its 

lateral edges being 6, and its basal edge 14. 

66. Find the volume of a frustum of a square pyramid, 

the edge of the lower base being 20, the edge of 
the upper base 10, and the altitude 12. 

67. A B C D EFGHisa rectangular parallelopiped, 

and I K is a diameter of the upper base. Pass a 
plane through I K and G H and determine the ra¬ 
tio of the remaining part to the part cut off. 

Define a Sphere: 

1 . As a solid. 

2 . By its generation. 

Define the Radius of a Sphere, the Diameter of a Sphere. 
When is a plane tangent to a sphere ? 

When is a polyhedron inscribed in a sphere? Circum¬ 
scribed about a sphere ? 

When is a sphere inscribed in a cylinder ? 

[68.] Every section of a sphere made by a cutting plane 
is a circle. 

Define a Great Circle. A Small Circle. 

[69.] To pass a circle through three points on the surface 
of a sphere: 

1 . How many small circles can be passed through two points 
on the surface of the sphere ? How many great circles. 


6 


GEOMETRICAL EXERCISES. 


2 . Under what conditions will three points on the surface of 
a sphere determine a great circle. 

Show: 

1. That every great circle bisects the sphere. 

2 . That any two great circles on the same sphere bisect each 

other. 

70. Any two points in the circumference of a circle of a 
sphere are equally distant from the pole of the 
circle. 

[71.] One sphere and but one can be circumscribed about 
any tetrahedron. 

[72.] A sphere may be inscribed in any tetrahedron. 

[73.] To find the diameter of a material sphere. 

[74.] If a straight line be resolved about another line in 
its plane as an axis, the area of the surface gener¬ 
ated is equal to the product of the projection of the 
revolving line upon the axis by the circumference 
whose radius is a perpendicular erected at the mid¬ 
dle point of the line, and terminated by the axis. 

[75.] Derive a formula for the area of the surface of a 
sphere in terms of R (its radius.) 

1. The surface of a sphere is equivalent to four great circles. 

2. The surface of a sphere is equivalent to a circle whose ra¬ 

dius is the diameter of the sphere. 

3. The surfaces of two spheres are to each other as the squares 

of their radii. 

Define a Spherical Sector: 

[76.] Derive a formula for the volume of a sphere in 
terms of R (its radius). 

1. Two spheres are to each other as the cubes of their radii. 

2. The volume of a spherical sector is equal to the product of 

the area of the base by one third of the radius of the 
sphere. 


GEOMETRICAL exercises. 


77 


77. A cylinder is circumscribed about a sphere; find the 

ratio: 

1. Of the lateral surface of the cylinder to the surface of the 

sphere. 

2. Of the entire surface of the cylinder to the surface of the 
sphere. 

V Of the volume of the cylinder to the volume of the sphere. 

78. A cube is circumsbribed about a sphere; find the 

ratio: 

1. Of the entire surface of the cube to the surface of the 

sphere. 

2. Of the volume of the cube to the volume of the sphere. 

79. The volume of a cube is 1000, find its radius. 

80. The surface of a sphere is 1256.64, find its volume. 

81. The volume of a sphere is 33510.4, find its surface. 

82. Find the diameter of a ball equivalent to the sum of 

three balls whose volumes are respectively 1, 2 and 3 

83. Given a sphere whose radius is R; to find the edge: 

1. Of the inscribed cube. 

2. Of the inscribed regular octahedron. 

3. Of the inscribed regular tetrahedron. 

84. In the sphere whose radius is R, find the ratio of the 

volume of the regular inscribed cube to the volume 
of the regular inscribed octahedron. 

85. A hollow ball is 10 feet in diameter. Find the volume 

of the solid portion if it has a uniform thickness*of 
two feet. 

86. If the earth’s diameter is 7,912 miles, find its surface 

in square miles. 

87. Find the volume of the earth in cubic yards. 

88. Giv.m a sphere whose radius is 10, to find the volume 

of an ungula formed by the intersection of two 
semicircles which meet at an angle of 6o°. 


78 


GEOMETRICAL EXERCISES. 


89. The depth of a hemispherical vessel is 14 inches. Find 

its capacity in liquid gallons. 

90. Find the volume of the earth’s atmosphere if its depth 

is 50 miles and the earth’s radius 4,000 miles. 

GEOMETRICAL FORMULAE. 

Abbreviations Employed. 

A. Altitude. 

B. Base, Right Section. 

b. Upper Base. 

E. Edge, Element. 

H. Slant Height. 

P. Perimeter of Right Section, Perimeter of Base, Perim¬ 
eter of Lower Base. 

p. Perimeter of Upper Base. 

R. Radius, Radius of Right Section, Radius of Base, 

Radius of Lower Base. 

r. Radius of Upper Base. 

S. Surface. 

V. Volume. 

The formulae for the pyramid and the frustum of a 
pyramid apply only to the right pyramid, and those for 
the cone and the frustum of a cone only to the cone of revo¬ 
lution. 

To the Teacher: The student who has completed 
Book VII. should be able to reproduce the following Table 
of Formulae, and should have a sufficient number of practic¬ 
al exercises to enable him to employ them readily. Before 
solving each exercise let him first write upon the board or 
manuscript the formula to be used. 


geometrical exercises. 


table of formulae 


i. Circle.* 

( Circumference, 2 - R. 

| Area, - R ! . 

\ Prism. j 

|Cylinder..,, -j 

:S., P E. 

IV., B E. 
i s., 2* RE. 

[ V., - R 2 E. 

| Pyramid.... j 
| Cone. ' 

r s., % ph. 

[ V., Yi B A. 
fS., - R H. 

[ V., }i - R 2 A. 


f Frustum of f S., ^(P+p)H. 

J a Pyramid j V., }i ACB+b+]/ Bb) 
4 * j Frustum of J S., - H(R-j-r). 

L a Cone. { V., /s * A(R+r+Rr). 

o i ( S., 4 - R 2 . 

5- Sphere. {v.,]-R 3 . 






80 


GEOMETRICAL EXERCISES. 


BOOK VIII. 

SPHERICAL POLYGONS. 

1. Define Zone, the Bases of a Zone, the Altitude of a 

Zone. 

2. What is a Spherical Segment? 

3. Define a Spherical Angle; Right, Oblique, Acute, Ob¬ 

tuse Spherical Angles. 

4. Define a Spherical Polygon. 

Designate by its name a spherical polygon of 3 sides; 
of 4 sides; of 5 sides; of 6 sides; of ysides; etc. 

5. Define spherical triangles, as Right, Oblique, Acute, 

Obtuse, Scalene, Isosceles, and Equilateral. 

6. What is the Polar of a Spherical Triangle? 

7. Define Symmetrical Spherical Triangles. 

[1.] If one spherical triangle is the polar of another, the 
second is the polar of the first. 

[2.] In two polar triangles each angle of either is the sup¬ 
plement of the side lying opposite to it in the other. 

1. The sides of a spherical triangle are respectively the sup¬ 

plements of the angles lying oppos.te to them in its polar 
triangle. 

2. The angles of a spherical triangle are respectively 80°, 

100°, and 150°; find the sides of its polar. 

3. The sides of a polar triangle are respectively 60°, 20°, and 

and 120°; find the angles of its polar. 

[3.] On the same spheres or on equal spheres, mutually 
equilateral triangles are mutually equiangular. 

[4.] State and prove the conversely. 

[5.] In an isosceles spherical triangle the angles oppo¬ 
site the equal sides are equal. 


GEOMETRICAL EXERCISES. 


81 


[6.] State and prove the conversely. 

[7.] The arc of a great circle drawn from the vertex of an 
isosceles spherical triangle to the middle point of 
the base : 

1 . Bisects the vertical angle. 

2. Is perpendicular to the base. 

3. Divides the triangle into two symmetrical triangles. 

[8.] Symmetrical spherical triangles are equivalent. 

[9.] Any side of a spherical triangle is less than the sum 

of the other two sides: 

1 . Any side of a spherical triangle is greater than the differ¬ 

ence of the other two sides. 

2 . Any side of a spherical polygon is less than the sum of 

the other sides. 

[10.] The sum of the angles of a spherical triangle is 
greater than two and less than six right angles. 

When is a spherical triangle Rectangular? Bi-rect- 
angular? Tri-rectangular ? 

What is meant by the Spherical Excess of a spheri¬ 
cal triangle? 

[11.] Find the ratio of a tri-rectangular triangle to the 
entire surface of the sphere on which it is placed. 

[12.] In any spherical triangle, the greater side is opposite 
the greater angle. 

[13.] In any spherical triangle, the greater angle is oppo¬ 
site the greater side. 

[14.] Test the equality of two spherical triangles having: 

1 . Two sides and the included angle of one equal respectively 

to two sides and the included angle of the other. 

2 . Two angles and the included side of one equal respectively 

to two angles and the included side of the other. 

Define a Tune; the Angle of a Eune; a Spherical 
Wedge. 


82 


GEOMETRICAL/ EXERCISES. 


[15.] A lune is to the surface of a sphere as the angle of 
the lune is to four right angles. 

[16.] If arcs of two great circles intersect upon the sur¬ 
face of a hemisphere, the sum of the opposite 
angles formed is equal to a lune whose angle is 
equal to the angle at which the circles intersect. 

[17.] The area of a spherical triangle is equal to its 
spherical excess multiplied by the tri-rectangular 
triangle. 

[18.] The area of a spherical polygon is equal to the 
spherical excess multiplied by the tri-rect¬ 
angular triangle. 

19. Find the area of a spherical triangle whose angles 

are 8o°, ioo° and 130°, if the radius of the sphere 
is 30 inches. 

20. Find the area of a spherical triangle upon a sphere 

whose radius is 9: 

1 . If the sum of its angles is 270 °. 

2 . If the sum of its angles is 360 °. 

[21.] The shortest distance between two points measured 
on the surface of a sphere is the arc of a great circle, 
not greater than a semi-circumference, which 
joins them. 


CxEOMETRICAL exercises. 


83 


supplementary exercises. 

The following exercises may be omitted at the discretion 
of the teacher, if considered too difficult for the 
class. 

Define Line of Centers, Center of Similitude, External 
Center of Similitude, Internal Center of Similitude. 

1. If in two unequal circles parallel radii be drawn the 

straight line joining their extremities will if pro¬ 
duced pass through the center of similitude. 

1. Through the external center of similitude if the radii are 

on the same side of the line of centers. 

2. Through the internal center of symmetry if the radii are 

on opposite sides of the line of centers. 

2. If from C (the center of similitude) of two unequal cir¬ 

cles a secant be drawn cutting the circumference 
consecutively in A B C and D, show that P AXP D 
=P BXP C. 

3. From the above propositions, derive a method of draw¬ 

ing a common tangent to two given circles. 

The following problems are known as the tangencies : 
Given two points A and B, two straight lines C and D and 
two circumferences E and F ; it is required to de¬ 
scribe a circumference: 

1. That shall pass through A and B and touch C: 

1. When C is parallel to the straight line A B. 

2. When C is oblique to the straight line A B. 

2. That shall pass through A and B and touch E. 

3 That shall pass through A and touch C at a given 
point. 

4. That shall pass through A and touch E at a given 

point. 


84 


geometrical exercises. 


5. That shall pass through A and touch C and D: 

1. When C and D are parallel. 

2. When C and D are convergent. 

6. That shall touch C and D and a third line G. 

7. That shall touch C, D and E: 

1. When C and D are parallel. 

2. When C and D are convergent. 

8. That shall pass through A and touch C and E. 

9. That shall touch C,E and F. 

10. That shall pass through A and touch E and F. 

11. That shall touch E and F and a third circumference IT. 


TABLE OF THE ENGLISH NUMERALS. 

Showing their corresponding terms in Anglo-Saxon, 
Latin and Greek. 


English. 

Anglo Saxon. 

Latin. 

Greek. 

One. 

An. 

Unus. 

Eis. 

Two. 

Twa. 

Duo. 

Duo. 

Three. 

Thry. 

Tres. 

Treis. 

Four. 

Feower. 

Quatuor. 

Tessares. 

Five. 

Fif. 

Quinque. 

Penta. 

Six. 

Seox. 

Sex. 

Hex. 

Seven. 

Seofan. 

Septem. 

Septa. 

Eight. 

Eahte. 

Octo. 

Octa. 

Nine. 

Nigon. 

Novem. 

Hennea. 

Ten. 

Tyn or tig. 

Decern. 

Deka. 

Eleven. 

Un-lif (one left.)Undecim. 

Hendeka. 

Twelve. 

Twa-lif. 

Duo-decim. Do-deka. 

Thirteen. 

Threotyn. 

Tri-decem. 

Tri-deka. 

Twenty. 

Twa tig. 

Viginta. 

Eikosa. 

One hundred. 

Hund. 

Centum. 

Hekaton. 

One thousand. 

Thusend. 

Mille. 

Kilioi. 


geometrical, exercises. 


85 


» 

VOCABULARY. 

Abbreviations : 

L,., Latin ; G., Greek; F., French; A. S., Anglo-Saxon. 

Adjacent Angles, (Ad and Jaceo).—Angles having a 
common vertex and a common side. 

Acute angle (L. Acutus, sharp).—An angle less than a, 
right angle. 

Acute triangle.—A triangle all of whose angles are acute* 

Alternation (L. Alternatio, change).—The interchanging 
of the position of the means of a proportion. 

Altitude (L. Altitudo, height).—The perpendicular dis¬ 
tance : 

1. Between the parallel sides of a trapezoid or a parallelogram; 

or 

2 . From the vertex to the base of a triangle ; or 

3 . Between the bases of a prism, the bases of a frustum of a pyra¬ 

mid or of a frustum of a cone ; or 

4 . From the apex to the base of a pyramid or a cone. 

Angle (Angulus.)—The difference in direction of two lines 
that meet in a point. 

Antecedent (L. Ante and cedere to go).—The first 
term of a ratio. 

Apex (L).—The point of meeting of the edges of a 
pryamid or of the elements of a cone. 

Apothem (G. Apo, upon, and tithami (to place).—The 
perpendicular from the center of a regular polygon 
to the middle point of one of its sides. 

Arc (L. Arcus).—Any part of the circumference of a 
circle. 

Area (L).—The surface of a geometrical magnitude of two 
or three dimensions. 

Axiom (G. Axioma, a principle).—A self evident truth. 


86 GEOMETRICAL EXERCISES. 

Axis (L).—The line about which the parts of a sym¬ 
metrical solid may be revolved. 

Basal Edge.—The intersection of the base of a prism or of 
a pyramid with one of its lateral faces. 

Base (E. basis).—The line or surface on which a geomet¬ 
rical magnitude is supposed to stand. 

Bi-reetangular.—Containing two right angles. 

Bisector (L. bis and secare).—A magnitude that divides 
another magnitude into two equal parts. 

Broken Line.—A succession of connected straight lines. 

Central Angle.—The angle formed by the meeting of two 
radii of a circle or of two radii of a regular polygon. 

Center (L. centrum).—A point equally distant from the 
extremities of a line, from the boundary of a polygon 
or circle, or from the surface of a solid. 

Center of Similitude.—The point of meeting of the com¬ 
mon tangent of two circles with their line of centers. 
When the common tangent is exterior, the point is 
called the External Center of Similitude. When the 
common tangent is interior, the point is called the 
Internal Center of Similitude. 

Chord (L. chorda, a string of a musical instrument).—A 
sraight line joining the extremities of an arc. 

Circle (L. circulus, from circus, a ring).—A plane surface 
bounded by a uniformly curved line. A regular 
polygon of an infinite number of sides. The surface 
generated by revolving a straight line about one of 
its extremities. 

Circumference (Circuin and ferre).—The bounding line of 
a circle. 

Circumscribe (Circum and scribere).—To place one magni- 


GEOMETRICAL EXERCISER 


87 


tude about another. A circle is circumscribed about 
a polygon when its circumference passes through 
each angle of the polygon. A polygon is circum¬ 
scribed about a circle, when each side of the polygon 
is tangent to the circle. A cylinder is circumscribed 
about a prism when its surfaces embrace each later¬ 
al edge of the prism. A prism is circumscribed about 
a cylinder when each of its lateral faces is tangent to 
the cylinder. In like manner a cone may be circum¬ 
scribed about a pyramid, or a pyramid about a cone. 
A sphere is circumscribed about a polyhedron when 
each of the vertices of the polyhedron lies in the sur¬ 
face of the sphere. A polyhedron is circumscribed 
about a sphere when each of its faces is tangent to 
the sphere. 

Commensurable (L,. con and mensurare, to measure).— 
Having a common measure. 

Complement of an Angle (I,, con and piere, to fill).—The 
difference between an angle and ninety degrees. 

Complementary Angles.—Two angles whose sum is nine¬ 
ty degrees. 

Composition (L. com and positium, placed). —The ar¬ 
rangement of the terms of a proportion, so that the 
sum of the first and second is to the first or second, as 
the sum of the third and fourth is to the third or 
fourth. 

Concave Polygon (L,. conandcavus, hollow).—A poly¬ 
gon having one of its angles re-entrant. 

Concentric (T- con and centrum, a center)—having a com¬ 
mon center.—The term is applied to circumferences, 
circles and spheres. 


88 


GEOMETRICAL EXERCISES. 


Cone (L. conus).—A solid bounded by a conical surface 
and a plane cutting all its elements. 

Cone of Revolution.—A solid generated by revolving a 
right triangle about one of its legs as an axis. 

Conical Surface.—A curved surface generated by the rev¬ 
olution of a straight line which in all its positions 
touches a given curve and passes through a fixed 
point without the curve. The curve is called the 
Directrix and the moving line the Generatrix. 

Consequent (L. con and sequi, to follow 7 ).—The second 
term of a ratio. 

Constant (L. con and stare, to stand).—A quantity whose 
value remains unchanged throughout a discussion. 

Construction (L. con and struere, to place).—The graphi¬ 
cal representation of the lines and figures necessary 
to the demonstration of a theorem or the solution of 
a problem. 

Convergent Lines (con and vergere).—Lines that will 
meet if sufficiently produced. 

Convex Polygon (L. convexus).—A polygon each of 
wdiose angles is less than two right angles. 

Corollary (L- corollarium, a corona).—A proposition 
readily deduced from the demonstration of a theorem. 

Couplet (F).—Either pair of ratios in a proportion. 

Cube (L. cubus).—A rectangular parallelopiped bounded 
by squares. 

Curved Line (L. curvus).—A line that changes its direc¬ 
tion at every point. The path of a moving point 
that continually changes its direction. 

Curved Surface.—The path of a moving straight line that 
is continually changing its direction. 


geometrical exercises. 


89 


Cylinder (L,. kulindron, to roll).—A solid bounded by a 
cylindrical surface and two parallel planes cutting all 
its elements. 

Cylinder of Revolution.—The solid generated by revolv¬ 
ing a rectangle about one of its sides as an axis. A 
regular right prism with an infinite number of lateral 
faces. 

Cylindrical Surface.—A curved surface generated by the 
revolution of a straight line which in all its positions 
touches a given curve and is parallel to another 
straight line outside the plane of the curved line. 
The curved line is called the Directrix, the revolving 
line the Generatrix. 

Decagon (G. deka and gonos).—A polygon of ten sides. 
Decreasing Variable.—A variable approaching its inferior 
limit. 

Demonstration (L,. de and monstrare, to show).—The pro¬ 
cess of proving a proposition. 

Diagonal (G. dia and gonos).—A straight line drawn 
froman angle of a polygon or a polyhedron to any 
other not adjacent angle. 

Diameter (G. dia and metron). 

1. Of a Circle. A chord passing through its centre. 

2. Of a Rectangle. A straight line joining the middle points 

of two opposite sides. 

3. Of a Sphere. A straight line passing through the center 

of the sphere and terminated by its surface. 

Dihedral Angle (G. dia and hedra)*— An angle formed by 
the intersection of two planes. 

Directrix (L. dirigere, to direct).—See Concave Surface, 
Cylindrical Surface. 

Division (L,. divisio).—The arrangement of the terms of a 


90 


GEOMETRICAL EXERCISES. 


proposition so that the difference of the first and sec¬ 
ond is to the first or second as the difference of the 
third and fourth is to the third or fourth. 

Dodekahedron (G. dodeka and hedra).—A polyhedron 
bounded by twelve faces. 

Edge (A. S. ecg).—The line in which two faces of a 
polyhedral angle or of a polyhedron meet. 

Equal Magnitudes (E. aequus).—Magnitudes having the 
same form and the same extension. The test of 
equality is superposition. 

Equiangular Polygons.—A polygon having all its angles 
equal. Two polygons are said to be mutually equi¬ 
angular when they have an equal number of sides 
and their corresponding angles are equal each to each. 

Equilateral Polygon (L. Aequus and latus).—A poly¬ 
gon having all its sides equal. 

Equilateral Triangle.—A triangle having all its sides 
equal. 

Equivalent Magnitudes (L. aequus, and valere, to be of 
value).—Magnitude having the same extension but 
unlike in form. 

Extension (E. extensio).—That property of a figure by 
which it occupies space. 

Exterior Angle of a Polygon (E. Exterior).—The angle 
formed by a side *of a polygon and the prolon¬ 
gation of the side adjacent. 

Extremes (E. extremis).—The first and fourth terms 
of a proportion. 

Face Angle of a Polyhedral Angle (E. facies).—The 
divergence of two edges of a polyhedral angle. 

Face of a Polyhedron.—One of its bounding polygons. 


geometrical exercises. 


91 


Figure (I,, figura).—The graphical representation of a* 
geometrical magnitude. 

Formula (L).—A mathematical proposition expressed 
in algebraic language. 

Fourth Proportional.—The final term in a proportion. 

Frustum (L).—The part of a pyramid or a cone inter¬ 
cepted between its base and a plane passed parallel to 
its base. 

Generation (L. generare, to produce).—The production 
of a geometrical magnitude by the movement in 
accordance with some mathematical law of another 
magnitude. 

Generatrix (I,, generare, to produce).—See Conical 
Surface, Cylindrical Surface. 

Geometry (G. ga, the earth, and metron).—That branch 
of mathematics which treats of the properties, rela¬ 
tions and extension of magnitudes. 

Great Circle.—The section of a sphere produced by pass¬ 
ing a plane through its center. 

Heptagon (G. Hepta, and gonos).—A polygon of seven 
sides. 

Hexagon (G. hexa and gonos).—A polygon of [six 
sides. 

Hexahedron (G. hexa, and liedra).—A polyhedron 
bounded by six faces. 

Homologous (G. homos, the same, and logos speech).— 
Corresponding in position and relation. 

Horizontal Line (G. horidzein, to bound).—A line 
parallel to the plane of the horizon. 

Hypotenuse (G. hupo, under, and teinein, to stretch).— 
The side of a right triangle opposite the right angle. 


92 


geometrical exercises. 


Icosahedron (G eikosa, and hedra).—A polyhedron 
bounded by twenty faces. 

Incommensurable (L. in, con and mensurare, to mea¬ 
sure).—Not having a common measure. 

Increasing Variable (L. in and crescere, to grow).—A vari¬ 
able approaching its superior limit. 

Inferior Limit of a Variable (L. inferior, lower).—See 
Limit. 

Inscribe (L. in and scribere).—To place one magnitude 
within another. A circle is inscribed in a polygon 
when its circumference touches each side of the 
polygon. A polygon is inscribed in a circle when 
each vertex of the polygon lies in the circumference 
of the circle. A cylinder is inscribed in a prism 
when its lateral surface is tangent to each side of the 
prism. A prism is inscribed in a cylinder when each 
lateral edge of the prism is an element of the cylin¬ 
der. In like manner a cone may be inscribed in a 
pyramid and a pyramid in a cone. A sphere is in¬ 
scribed in a polyhedron when its surface is tangent 
to each face of the polyhedron. A polyhedron is in¬ 
scribed in a sphere when each of its vertices lies in 
the surface of the sphere. 

Intersection (L. inter, between, and secare).—The point of 
meeting of two lines. The line in which two planes 
meet. 

Inversion (L.in, and vertere).—The placing, in a propor¬ 
tion, of tne antecedents for the consequents and the 
consequents fcr the antecedents. 

Isosceles Trapezoid (G. isos, equal, and skelos, leg).—A 
trapezoid having its non-parallel sides equal. 


geometrical exercises. 


93 


Isosceles Triangle.—A triangle having two of its sides 
equal. 

Isosceles Trihedral Angle.—A trihedral angle having two 
of its face angles equal. 

Lateral Edge (L. latus, a side).—The intersection of 
two adjacent lateral sides of a prism, ol a pyramid, or 
of a frustum of a pyramid. 

Lateral Face.—One of the faces about the sides of a prism, 
of a pyramid, or of a frustum of a pyramid. 

Lateral Surface.—The sum of the lateral faces of a prism, 
of a pyramid, or of a frustum of a pyramid. The curv¬ 
ed surface of a cylinder, of a cone, or of a frustum of a 
cone. 

Leg.—One of the perpendicular sides of a right triangle. 
The term is sometimes applied to any side of a trian¬ 
gle. 

Lemma (L.).—A propostion demonstrated as a pre¬ 
liminary to a succeeding demonstration. 

Limit (L. limis).—A value to which a variable through¬ 
out a discussion may approach indefinitely near, 
but which it can not attain. The limit which an 
increasing variable thus approximates is called the 
Superior Limit of the variable. In like manner a 
decreasing variable approximates its Inferior Limit. 

Line (L. linea, a linen thread).—Extension in one direc¬ 
tion only. The path of a moving point. A line is 
sometimes regarded as a surface having zero for one 
of its dimensions, or as a solid having zero for its 
breadth and thickness. 

Line of Centers.—The straight line determined by the 
centers of two circles. 


94 


GEOMETRICAL EXERCISES. 


Locus of a Point (L. locus).—A line or collection of lines 
in a plane containing all the points that possess a 
common and determined property. 

Locus of a Point in Space.—A surface or collection of 
surfaces containing all the points that possess a com¬ 
mon and determined property. 

Lune (L- luna, the moon).—A portion of the surface of a 
sphere bounded by two semi-circumferences. 

Magnitude (L. magnitudo).—That which has one or more 
of the three dimensions. 

Mean Proportional (L. medianus).—A quantity equal to 
the square root of the product of two other quantities. 

Means.—The second and third terms of a proportion. 

Median of a Triangle (L. medianus).—A straight line 
from #ny vertex of a triangle to the middle point of 
the opposite side. 

Median of a Trapezoid.—A straight line joining the mid¬ 
dle points of the non-parallel sides of a trapezoid. 

Multiple (L. multus and plicare, to fold).—A quantity 
that will contain another quantity an integral num¬ 
ber of times without a remainder. 

Oblique Angle (L. obliquus).—An angle formed by the 
meeting of two lines not perpendicular to each other. 

Oblique Line.—A line which will meet another line if 
sufficiently produced. 

Oblique Triangle.—A triangle having all its angles 
oblique. 

Oblique Parallelopiped.—A parallelopiped whose lateral 
edges are oblique to the planes of its bases. 

Oblique Prism.—A prism whose lateral edges are oblique 
to the planes of its bases. 


geometrical exercises. 


95 


Obtuse Triangle.—A triangle one of whose angles is 
greater than a right angle. 

Oblong (L. ob, and longus).—A rectangle having its adja¬ 
cent sides unequal. 

Octagon (G. octo and gonos).—A polygon of eight sides. 

Octahedron (G. octo and hedra).—A polyhedron bounded 
by eight faces. 

Opposite Angle (L, ob and positus, placed).—See Verti¬ 
cal Angle. 

Parallel Lines (G. para and allelon, another).—Lines that 
are everywhere equally distant from each other. 

Parallelogram (G. para, allelon, and gramma, aline)—A 
quadrilateral whose opposite sides are parallel. 

Parallelopiped or Parallelopipedon (G. para, allelon and 
pipedos, level).—A prism having a parallelogram for 
its base. 

Parallel Planes.—Two planes that are everywhere equally 
distant from each other. 

Pentagon (G. penta and gonos).—A polygon of five 
sides. 

Perimeter (G. peri, around, and metron, a measure).— 
The broken line bounding a polygon. 

Perpendicular Line (L. per and pendere, to hang down).— 
A line meeting another line so as to be inclined to it 
neither to the right nor the left. 

Plane (L* planus).—A surface such that a straight line 
joining any two points within it lies wholly in the 
surface. The surface generated by moving a straight 
line continuously in the same direction. 

Plane Angle of a Dihedral Angle.—The angle formed by 
two straight lines drawn one in each face of the dih e ~ 


96 


geometrical exercises. 


dral angle and perpendicular to its edge at the same 
point. 

Plane Surface.—A surface that does not change its 
direction however far produced. 

Point (L. punctum, from pungere, to pierce).—Position 
without magnitude. A point is sometimes regarded 
as a line having zero for its length; or as a surface 
having zero for its length and breadth; or as a solid 
having zero for each of its three dimensions. 

Polar Triangles (G. polus).—Two spherical triangles so 
related each to each that the vertices of one are the 
poles of the sides of the other. 

Pole (G. polus).—A point in the surface of a sphere 
equally distant from every point in the circumference 
of a great circle. 

Polygon (G. polus and gonos).—A plane surface bounded 
by straight lines. 

Polyhedron (G. polus and hedra).—A solid bound by 
plane surfaces. 

Polyhedral Angle.—An angle formed by three or more 
plane angles which meet at a point. 

Postulate (L- postulare, to demand).—A proposition stat¬ 
ing a self-evident possibility. 

Prism (G. prisma, from pridzein, to saw).—A polyhedron 
having equal and parallel polygons for bases and 
parallelograms for its lateral sides. 

Problem (G. pro and ballein, to throw).—A proposition 
requiring solution. 

Projection (L. pro andjacere). 

1. Of a point on a line or a plane. The foot of a perpen¬ 

dicular from the point to the line or plane. 

2. Of a line on another line or on a plane. The locus in the 


GEOMETRICAL EXERCISES. 


97 


line or plane of the projection of all the points in the 
given line. 

Proportion (I*, pro and portio, a part).—An equality of 
ratios. 

Proposition (T. pro and ponere).—Any formal mathemat¬ 
ical statement. 

Pyramid (G. pur, a flame, and eidos, shape).—A polyhe¬ 
dron whose base is a polygon and whose lateral sides 
are triangles which meet in a common vertex. 
Quadrangular (Iy. quatuor andangulus).—Having four 
angles. 

Quadrant (L,. quadrans, a fourth part).—A sector whose 
arc is ninety degrees. 

Quadrilateral (X. quatuor and latus).—A polygon of four 
sides. 

Radius (L. radius, a ray). 

1. Of a Circle. A straight line from the center of a circle to 

any point in the circumference. 

2 . Of a Regular Polygon. A straight line from the center of 

the circumscribing circle to any vertex of the polygon. 

3. Of a Sphere. A straight line from the center of the sphere 

to any point in the surface. 

Ratio (L. ratio, calculation).—The quotient arising from 
dividing one magnitude by another. 

Rectangle (L. rectus, right, and angulus).—A parallelo¬ 
gram, all of whose angles are right angles. 
Rectangular.—Containing a right angle. 

Re-entrant Angle (L. re and inter).—An angle of a 
polygon projecting within its surface. 

Regular Polygon (L. regula, a rule).—A polygon that is 
both equiangular and equilateral. 

Regular Polyhedron.—A polyhedron bounded by regular 
polygons and having all its polyhedral angles equal. 


98 


GEOMETRICAL EXERCISES. 


Regular Prism.—A right prism whose bases are regular 
polygons. 

Regular Pyramid.—A pyramid whose base is a regular 
polygon, and whose vertex is in the perpendicular 
to the base at the center. 

Rhomboid (G. rhombus, a rhomb, andeidos, shape).—An 
oblique angled parallelogram. 

Rhombus or Rhomb (G. rhombus).—An equilateral 
rhomboid. 

Right Angle (L,. rectus).—An angle of ninety degrees. 
The angle formed by the meeting of two perpendicu¬ 
lar lines. 

Right Section.—A section of a prism, pyramid, cylinder, 
cone, or lrustum, made by passing a plane perpendic¬ 
ular to its edges or elements. 

Right Prism.—A prism, the plane of whose lateral faces 
are perpendicular to its base. 

Scalene Trapezoid (G. skalanus, unequal).—A trapezoid 
whose non-parallel sides are unequal. 

Scalene Triangle.—A triangle having no two sides equal. 

Scalene Tri-hedral Angle.—A tri-hedral angle having no 
two of its face angles equal. 

Scholium (L.).—A remark subjoined to a demonstra¬ 
tion. 

Secant (E. secare).—A straight line passing through a 
circle and cutting its circumference in two points. 

Secant Plane.—A plane cutting a solid. 

Sector (E. secare).—That part of a circle bounded by two 
radii and the arc they intercept. 

Segment (E- segmentum).—That part of a circle which is 
bounded by a chord and the arc which it subtends. 


geometrical exercises. 


99 


Semicircle (L,. semi, half).—One of the equal parts into 
which the diameter divides the circle. 

Similar Polygons (L. Similis).—Polygons having their 
angles respectively equal each to each and the cor¬ 
responding sides proportional. 

Remark.—In different circles two arcs, two sectors, or two 
segments having equal central angles are similar. 

Similar Solids.—Solids bounded by the same number of 
similar polygons similarly placed and having their 
corresponding polyhedral angles equal. 

Slant Height.—The altitude of any lateral face of a 
regular pyramid or frustum of a regular pyramid. 
Any element of a cone of revolution or of a 
frustum of a cone of revolution. 

Small Circle.—Any section of a sphere made by a plane 
not passing through its center. 

Solid (L,. solidus).—A magnitude of three dimensions. A 
limited portion of space. The path of a moving sur¬ 
face. 

Sphere (L. sphera).—A solid bounded by a uniformly 
curved surface. A solid generated by the revolution 
of a semicircle about the bounding diameter as an 
axis. 

Spherical Angle.—An angle formed by the intersection 
upon a sphere of two arcs of a great circle. 

Spherical Excess.—The excess of the sum of the angles 
of a spherical triangle over two right angles. 

Spherical Polygon.—A portion of the surface of sphere 
bounded by arcs of great circles. 

Spherical Sector.—A portion of a sphere generated by 
the revolution of a sector of a great circle about 
one of its radii as an axis. 


100 


GEOMETRICAL EXERCISES. 


Spherical Segment.—A portion of a sphere included be¬ 
tween two parallel planes. 

Spherical Triangle. —A spherical , polygon of three 
sides. 

Square (L. ex and quatuor).—An equilateral rectangle. 

Straight Line (A. S., streht, to stretch).—A line that does 
not change its direction. The shortest distance be¬ 
tween two points. 

Straight Angle.—An angle of one hundred and eighty 
degrees. 

Sub-Multiple.—A number that is contained in another 
number an integral number of times. 

Subtend (L. sub and tendere, to stretch).—To extend 
under. A central angle subtends the arc inter¬ 
cepted by the radii which form its sides. A chord 
subtends the arc which it intercepts. 

Superior Limit of a Variable (L- superior, comparative 
of superus, being above).—See Limit. 

vSuperposition (L. super, and ponere).—The placing of one 
magnitude upon another to test their equality. ‘ 

Supplement of an Angle (L- sub and plere, to fill).—The 
difference between an angle and i8o°. 

Supplementary Adjacent Angles.—Two supplementary 
angles which have a common vertex and a common 
side. 

Surface (L. super and facies, a face).—A magnitude of 
two dimensions. The path of a moving straight 
line. A surface is sometimes regarded a solid, 
having zero for its thickness. 

Symmetrical Solid.—(See Symmetry.) 

Symmetrical Surface.—(See Symmetry.) 


GEOMETRICAL EXERCISES. 


101 


Symmetry (G. sun and metron).—Such an arrangement 
of two geometrical magnitudes or of the parts of a 
geometrical magnitude with reference to a center 
that any point in the one has a corresponding point 
in the other on the opposite side of the center and 
equidistant with it from the center. A surface that 
can be divided into two equal parts that are thus re¬ 
lated is called a symmetrical surface. A solid that 
can be so divided is called a symmetrical solid. The 
center of symmetry may be a point, an axis (line), or 
a plane. 

I. Two points are symmetrical: 

1. With reference to a point when the straight line connect¬ 

ing them is bisected by that point. 

2. With reference to an axis when the straight line connect¬ 

ing is then perpendicular to the axis and is bisected by it. 

3. With reference to a plane ; when the straight line con¬ 

necting them is perpendicular to the plane and bisected 
by it. 

II. Two lines are symmetrical: 

1. With reference to a point when every straight line termi¬ 

nated by the lines and passing through the point is 
bisected by it. 

2. With reference to an axis: 

When every straight line terminated by the lines and 
drawn perpendicular to the axis is bisected by it. 

3. With reference to a plane. 

When every straight line terminated by the lines and 
drawn perpendicular to the axis is bisected by it. 

III. Two planes are symmetrical: 

1. With reference to a point. 

When every straight line terminated by the planes and 
drawn through the point is bisected by it. 

2. With reference to an axis. 


102 


geometrical exercises. 


When every straight line terminated by the planes and 
drawn perpendicular to the axis is bisected by it. 

3. With reference to a plane. 

When every straight line terminated by the two planes 
and drawn perpendicular to the third plane is bisected 
by it. 

Symmetry with reference to a point is sometimes called 
Perfect Symmetr}", to distinguish it from symmetry 
with reference to an axis or a plane, or Relative Sym- 
metery. 

Tangent (L. tangere, to touch).—A straight line which 
has one and but one point in common with the cir¬ 
cumference of a circle which lies in the same plane. 
A tangent to a circle is technically said to “touch” 
the circumference. 

Tangent Plane.—A plane which has the property of 
touching without intersecting a curved surface. A 
plane is tangent to a cjdindrical or conical surface 
when it embraces one and but one element of the sur¬ 
face. A plane is tangent to a sphere, when one and 
but one point in the surface of the sphere lies with¬ 
in the plane. 

Tetrahedron (G. tetares and hedra).—A polyhedron 
bounded by four faces. 

Theorem (G. theorama).—A proposition to be demon¬ 
strated. 

Third Proportional. —The fourth term in a proportion 
containing a mean proportional. 

Touch.—See Tangent. 

Transversal (L. trans and vertere).—A straight line inter¬ 
secting two or more other straight lines. 

Trapezium (L. trapezion).—A quadrilateral having no 
tw T o sides parallel. 


GEOMETRICAL, EXERCISES. 


103 


Trapezoid (G. trapezion and eidos, like).—A quadrilater¬ 
al having only two sides parallel. 

Triangle (E. tres and angulus).—A polygon of three 
sides. 

Triangular Prism.—A prism having a triangle for its 
base. 

Triangular Pyramid.—A pyramid having a triangle for 
its base. 

Trigon (G. tres and gonos).—A polygon of three sides. 

Trihedral Angle (G. tres and hedra).—A polyhedral 
angle formed by three planes. 

Tri-rectangular.—Containing three right angles. 

Truncated Prism, Pyramid, Cone, or Cylinder (E. trun- 
care, to mutilate).—That part of a magnitude 
included between its base and a plane not parallel to 
the base, passed through its edges or elements. 

Ungula (L,. ungula, a claw).—A spherical wedge. 

Unit of Measure (E. unus).—A standard of comparison 
used in measuring magnitudes of the same kind. 

Variable (E. variabilis).—A quantity whose value is con¬ 
stantly changing throughout a discussion. 

Vertex (E).—The point of meeting of any two sides of a 
polygon. In the triangle the term is often specially 
applied to the point of meeting of the sides of the 
angle opposite the base. 

Vertical Angle (E. vertex).—The angle included by the 
sides produced of another angle. Vertical angles are 
often called Opposite Angles. 

Vertical Eine.—Aline perpendicular to the plane of the 
horizon. 

Zero (F).—A quantity less than any assignable quantity. 


104 


geometrical exercises. 


Zone (G. zona).—The portion of the surface of a sphere 
intersected between two secant ‘planes. 

GREEK AND EATIN ROOTS AND AFFIXES 

Frequently appearing in geometrical terms. 

(G. Greek; L. Latin.) 


Ad, L. To. 

Angulus, E. An angle. 
Ante, E. Before. 

Bis, L. Twice. 

Circum, E. Around. 
Con, E- Together, with. 
De, L. Down. 

Dia, G. Through. 

Dis or di, G. Twice. 

E, E., Out of. 

Ex, E. From. 

Ferre, E. To bear. 
Gonos, G. An angle. 
Hedra, G. A side. 

In, E. In. 

Jacere, E. To lie. 

Eatus, E- A side. 
Eongus, E. Eong. 


Medianus, E. Middle. 
Magnus, E. Great. 

Metron, G. A measure. 
Multus, E. Much. 

Ob, E. Against. 

Oid, from eidos, G. Shape. 
Para, G. Beside. 

Ponere, E- To place. 

Poly, from polus, G. Many. 
Pro, E- Before. 

Scribere, E. To write. 
Secare, E. To cut. 

Sub, E. Under. 

Super, E. Over. 

Sun or Syn, G. With. 
Trans, E. Across. 

Vertere, E. To turn. 





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